WO1997014105A9 - An expert system for testing industrial processes and determining sensor status - Google Patents

Abstract

A method and system (110) for monitoring both an industrial process and a sensor (104). The method and system include determining a minimum number of sensor pairs needed to test the industrial process as well as the sensor (104) for evaluating the state of operation of both. After obtaining two signals associated with one physical variable, a difference function is obtained by determining the arithmetic difference between the pair of signals over time. A frequency domain transformation is made of the difference function to obtain Fourier modes describing a composite function. A residual function is obtained by subtracting the composite function from the difference function and the residual function (free of nonwhite noise) is analyzed by a statistical probability ratio test.

Description

An Expert System for Testing Industrial Processes and Determining Sensor Status

The United States Government has rights in this invention pursuant to Contract W-31-109-ENG-38 between the U.S. Department of Energy and the University of Chicago.

The present invention is concerned generally with an expert system and method for reliably monitoring industrial processes using a set of sensors. More particularly, the invention is concerned with an expert system and method for establishing a network of industrial sensors for parallel monitoring of industrial devices. The expert system includes a network of highly sensitive pattern recognition modules for automated parameter surveillance using a sequential probability ratio test.

Conventional parameter-surveillance schemes are sensitive only to gross changes in the mean value of a process, or to large steps or spikes that exceed some threshold limit check. These conventional methods suffer from either large numbers of false alarms (if thresholds are set too close to normal operating levels) or a large number of missed (or delayed) alarms (if the thresholds are set too expansively). Moreover, most conventional methods cannot perceive the onset of a process disturbance or sensor deviation which gives rise to a signal below the threshold level for an alarm condition.

In another conventional monitoring method, the Sequential Probability Ratio Test ("SPRT") has found wide application as a signal validation tool in the nuclear reactor industry. Two features of the SPRT technique make it attractive for parameter surveillance and fault detection: (1) early annunciation of the onset of a disturbance in noisy process variables, and (2) the SPRT technique has user-specifiable false-alarm and missed-alarm probabilities. One important drawback of the SPRT technique that has limited its adaptation to a broader range of applications is the fact that its mathematical formalism is founded upon an assumption that the signals it is monitoring are purely Gaussian, independent (white noise) random variables. It is therefore an object of the invention to provide an improved method and system for continuous evaluation and/or modification of industrial processes and/or sensors monitoring the processes.

It is also an object of the invention to provide an improved method and system for automatically configuring a set of sensors to monitor an industrial process.

It is another object of the invention to provide a novel method and system for statistically processing industrial process signals having virtually any form of noise signal.

It is a further object of the invention to provide an improved method and system employing identical pairs of sensors for obtaining redundant readings of physical processes from an industrial process.

It is still an additional object of the invention to provide a novel method and system utilizing a plurality of signal pairs to generate difference functions to be analyzed for alarm information.

It is still a further object of the invention to provide an improved method and system including a plurality of single sensors for each industrial device or process for providing a real signal characteristic of a process and further providing a predicted sensor signal allowing formation of a difference signal between the predicted and real signal for subsequent analysis.

It is also an object of the invention to provide a novel method and system wherein difference functions are formed from pairs of sensor signals operating in parallel to monitor a plurality of like industrial devices.

It is yet an additional object of the invention to provide an improved method and system utilizing variable and multiple pairs of sensors for determining both sensor degradation and industrial process status.

It is still another object of the invention to provide a novel method and system having a plurality of sensors analytically configured by an expert system to establish the minimum array of coupled sensors needed to monitor each sensor as well as the industrial process or devices.

An expert system has been developed that continuously monitors digitized signals from a set of sensors which are measuring a variety of physical variables (e.g., temperature, pressure, radiation level, vibration level, etc.). The expert system employs a sensitive pattern-recognition technique, the sequential probability ratio test ("SPRT") technique for early annunciation of sensor operability degradation. A SPRT module can monitor output from two identical sensors and determine if the statistical quality of the noise associated with either signal begins to change. In applications involving two or more industrial devices operated in parallel and equipped with identical sensors, a SPRT module applied to pairs of sensors monitoring the same physical process on the respective devices will provide sensitive annunciation of any physical disturbance affecting one of the devices. If each industrial device had only one sensor, it would not be possible for the SPRT technique to distinguish between equipment degradation and degradation of the sensor itself. In this application the primary benefit of the SPRT method would derive from its very early annunciation of the onset of the disturbance. Having this valuable information, additional diagnosis can then be performed to check the performance and calibration status of the sensor and to identify the root-cause of the signal anomaly.

For cases where each industrial device is equipped with multiple, redundant sensors, one can apply SPRT modules to pairs of sensors on each individual device for sensor- operability verification. In this case the expert system provides not only early annunciation of the onset of a disturbance, but also can distinguish between equipment degradation and degradation of its sensors. Moreover, when the expert system determines that the cause of the discrepant signals is due to a degraded sensor, it can identify the specific sensor that has failed.

In a simple generic application involving a single industrial device equipped with triply-redundant sensors for measurement of two physical variables, the expert system first identifies the minimum unique set of signal pairs that will be needed for the network of interacting SPRT modules. Further, the system can operate using two industrial devices working in parallel (ej^, jet engines, propeller drive motors on a ship, tøbomachinery in an industrial plant, etc.). Again the expert system identifies the pair-wise sensor combinations that it uses subsequently in building the conditional branching hierarchy for the SPRT -module configuration.

Other objects, features, alternative forms and advantages of the present invention will be readily apparent from the following description of the preferred embodiments thereof, taken in conjunction with the accompanying drawings described below.

Brief Description of the Drawings

FIGURE 1 illustrates the specified output of a pump's power output over time;

FIGURE 2 shows a Fourier composite curve generated using the pump spectral output of FIG. 1;

FIGURE 3 illustrates a residual function characteristic of the difference between FIGS. 1 and 2;

FIGURE 4A shows a periodogram of the spectral data of FIG. 1 and FIG. 4B shows a periodogram of the residual function of FIG. 3;

FIGURE 5 A illustrates a noise histogram for the pump power output of FIG. 1 and FIG. 5B illustrates a noise histogram for the residual function of FIG. 3;

FIGURE 6A shows an unmodified delayed neutron detector signal from a first sensor and FIG. 6B is for a second neutron sensor; FIG. 6C shows a difference function characteristic of the difference between data in FIG. 6A and 6B and FIG. 6D shows the data output from a SPRT analysis with alarm conditions indicated by the diamond symbols;

FIGURE 7A illustrates an unmodified delayed neutron detector signal from a first sensor and FIG. 7B is for a second neutron sensor; FIG. 7C shows a difference function for the difference between the data of FIG. 7A and 7B and FIG. 7D shows the result of using the instant invention to modify the difference function to provide data free of serially correlated noise to the SPRT analysis to generate alarm information and with alarm conditions indicated by the diamond signals;

FIGURE 8 A and B illustrate a schematic functional flow diagram of the invention with FIG. 8A showing a first phase of the method of the invention and FIG. 8B shows the application of the method of the invention;

FIGURE 9 illustrates a plurality of sensors monitoring two physical variables of a single industrial device;

FIGURE 10 illustrates triply -redundant sensors monitoring two physical variables for two industrial devices;

FIGURE 11 illustrates an overall system structure employing the sensor array of FIG. 10;

FIGURE 12 illustrates triply -redundant sensors monitoring one physical variable for three industrial devices;

FIGURE 13 illustrates a logic diagram and conditional branching structure for an equipment surveillance module; and

FIGURE 14 illustrates a system of industrial devices and development of SPRT modules for monitoring the system.

Detailed Description of Preferred Embodiments

In a method of the invention signals from industrial process sensors can be used to annunciate, modify or terminate degrading or anomalous processes. The sensor signals are manipulated to provide input data to a statistical analysis technique, such as a process entitled Sequential Probability Ratio Test ("SPRT"). Details of this process and the invention therein are disclosed in S.N. 07/827,776 which is incorporated by reference herein in its entirety. A further illustration of the use of SPRT for analysis of data bases is set forth in U.S. Pat. No. 5,410,422 and copending application of the assignee SN 08/068,713, also incorporated by reference herein in their entirety. The procedures followed in a preferred methods are shown generally in FIG. 8 and also in FIG. 12. In performing such a preferred analysis of the sensor signals, a dual transformation method is performed, insofar as it entails both a frequency-domain transformation of the original time-series data and a subsequent time-domain transformation of the resultant data. The data stream that passes through the dual frequency-domain, time-domain transformation is then processed with the SPRT procedure, which uses a log-likelihood ratio test. A computer software package, Appendix A, is also attached hereto covering the SPRT procedure and its implementation in the context of, and modified by, the instant invention.

In one preferred embodiment, successive data observations are performed on a discrete process Y, which represents a comparison of the stochastic components of physical processes monitored by a sensor, and most preferably pairs of sensors. In practice, the Y function is obtained by simply differencing the digitized signals from two respective sensors. Let y_k represent a sample from the process Y at time t_k During normal operation with an undegraded physical system and with sensors that are functioning within specifications the y_k should be normally distributed with mean of zero. Note that if the two signals being compared do not have the same nominal mean values (due, for example, to differences in calibration), then the input signals will be pre-normalized to the same nominal mean values during initial operation.

In performing the monitoring of industrial processes, the system's purpose is to declare a first system, a second system, etc., degraded if the drift in Y is sufficiently large that the sequence of observations appears to be distributed about a mean +M or -M, where M is our pre-assigned system-disturbance magnitude. We would like to devise a quantitative framework that enables us to decide between two hypotheses, namely:

H, : Y is drawn from a Gaussian probability distribution function ("PDF") with mean

M and variance σ².

H₂ : Y is drawn from a Gaussian PDF with mean 0 and variance σ². We will suppose that if H, or H₂ is true, we wish to decide for H, or H₂ with probability (1 - β) or (1 - α), respectively, where α and β represent the error (misidentification) probabilities.

From the conventional, well known theory of Wald, the test depends on the likelihood ratio l_n, where

The probability of observed sequence y,, y₂..., y_n given H, true \_ - (1)

The probability of observed sequence y„ y₂..., y_n given H₂ true After "n" observations have been made, the sequential probability ratio is just the product of the probability ratios for each step: l_n = (PR₁ (PR₂).....(PR (2) or i . f(y.|H.) ^l _n ⁼ π ⁽³⁾ _l=l f(y.|H where f(y|H) is the distribution of the random variable y.

Wald's theory operates as follows: Continue sampling as long as A<l_n<B. Stop sampling and decide H, as soon as l_n≥B, and stop sampling and decide H₂ as soon as l_n≤A.

The acceptance thresholds are related to the error (misidentification) probabilities by the following expressions:

^Λ - ^β -, and B = --P- (4)

1- a a

The (user specified) value of α is the probability of accepting H_j when H₂ is true (false alarm probability), β is the probability of accepting H₂ when H, is true (missed alarm probability). If we can assume that the random variable y_k is normally distributed, then the likelihood that H_j is true (i.e., mean M, variance σ²) is given by:

L( „ ,.., H₁) = ^ _nex_P[-^₂(∑_>;-2∑_yιM₊∑ M^!)] (5) k = l k = l k=l

Similarly for H₂ (mean 0, variance σ²):

L(y., y₂..., y_n IH₂) = (⁶)

The ratio of (5) and (6) gives the likelihood ratio l_n

l_n = exp[-^-₂ M(M-2y_k)] (7) k-l

Combining (4) and (7), and taking natural logs gives

ln- -<^-₂ T M(M-2y)<ln^- (8) k-l

Our sequential sampling and decision strategy can be concisely represented as:

Ifl_n≤ln- -, Accept H₂ (9)

1-α

Ifln-^-<l_n<ln-^, Continue Sampling (10)

\- a a

Andifl_n>ln^, Accept H, (11) a

Following Wald's sequential analysis, it is conventional that a decision test based on the log likelihood ratio has an optimal property; that is, for given probabilities α and β there is no other procedure with at least as low error probabilities or expected risk and with shorter length average sampling time.

A primary limitation that has heretofore precluded the applicability of Wald-type binary hypothesis tests for sensor and equipment surveillance strategies lies in the primary assumption upon which Wald's theory is predicated; l__, that the original process Y is strictly "white" noise, independently-distributed random data. White noise is thus well known to be a signal which is uncorrelated. Such white noise can, for example, include Gaussian noise. It is, however, very rare to find physical process variables associated with operating machinery that are not contaminated with serially-correlated, deterministic noise components. Serially correlated noise components are conventionally known to be signal data whose successive time point values are dependent on one another. Noise components include, for example, auto-correlated (also known as serially correlated) noise and Markov dependent noise. Auto-correlated noise is a known form of noise wherein pairs of correlation coefficients describe the time series correlation of various data signal values along the time series of data. That is, the data Ui , U2, . . ., U_n have correlation coefficients (Uι , U2), (U2, U3), . . ., (U_n_ I, U_n) and likewise have correlation coefficients (Ui , U3) (U2, U4), etc. If these data are auto-correlated, at least some of the coefficients are non-zero. Markov dependent noise, on the other hand, is a very special form of correlation between past and future data signals. Rather, given the value of U^, the values of U_n, n > k, do not depend on the values of Uj where j < k. This implies the correlation pairs (Uj, U_n), given the value Ufc, are all zero. If, however, the present value is imprecise, then the correlation coefficients may be nonzero. One form of this invention can overcome this limitation to conventional surveillance strategies by integrating the Wald sequential-test approach with a new dual transformation technique. This symbiotic combination of frequency-domain transformations and time- domain transformations produces a tractable solution to a particularly difficult problem that has plagued signal-processing specialists for many years.

In one preferred embodiment of the method shown in detail in FIG. 8, serially- correlated data signals from an industrial process can be rendered amenable to the SPRT testing methodology described hereinbefore. This is preferably done by performing a frequency-domain transformation of the original difference function Y. A particularly preferred method of such a frequency transformation is accomplished by generating a Fourier series using a set of highest "1" number of modes. Other procedures for rendering the data amenable to SPRT methods includes, for example, auto regressive techniques, which can accomplish substantially similar results described herein for Fourier analysis. In the preferred approach of Fourier analysis to determine the "1" highest modes (see FIG. 8A):

N

^Yt ⁼ f ⁺ Σ <^a" ^{cos ω} t + b_m sin ω_mt) (12) m=l where a₀/2 is the mean value of the series, a_m and b_m are the Fourier coefficients corresponding to the Fourier frequency ω_m, and N is the total number of observations. Using the Fourier coefficients, we next generate a composite function, X^ using the values of the largest harmonics identified in the Fourier transformation of Y_t. The following numerical approximation to the Fourier transform is useful in determining the Fourier coefficients a_m and b_m. Let X_j be the value of X, at the jth time increment. Then assuming 2 π periodicity and letting ω_m = 2πm/N, the approximation to the Fourier transform yields:

2 ^N'' 2 ^N"' ^a- ⁼ N Σ j=0 ^XJ ^{C0S ω}- ^{J bm =} N Σ j-0 ^{Xj S l} ^ ⁽¹³⁾ for 0 <m <N/2. Furthermore, the power spectral density ("PSD") function for the signal is given by l_m where a² _m + b² _m l_a = N (14)

2

To keep the signal bandwidth as narrow as possible without distorting the PSD, no spectral windows or smoothing are used in our implementation of the frequency-domain transformation. In analysis of a pumping system of the EBR-II reactor of Argonne National Laboratory, the Fourier modes corresponding to the eight highest l_m provide the amplitudes and frequencies contained in X_t. In our investigations for the particular pumping system data taken, the highest eight l_m modes were found to give an accurate reconstruction of X, while reducing most of the serial correlation for the physical variables studied. In other industrial processes, the analysis could result in more or fewer modes being needed to accurately construct the functional behavior of a composite curve. Therefore, the number of modes used is a variable which is iterated to minimize the degree of nonwhite noise for any given application. As noted in FIG. 8A a variety of noise tests are applied in order to remove serially correlated noise.

The reconstruction of X_t uses the general form of Eqn. (12), where the coefficients and frequencies employed are those associated with the eight highest PSD values. This yields a Fourier composite curve (see end of flowchart in FIG. 8A) with essentially the same correlation structure and the same mean as Y_t. Finally, we generate a discrete residual function P^ by differencing corresponding values of Y_t and X^. This residual function, which is substantially devoid of serially correlated contamination, is then processed with the SPRT technique described hereinbefore.

In a specific example application of the above referenced methodology, certain variables were monitored from the Argonne National Laboratory reactor EBR-II. In particular, EBR-II reactor coolant pumps (RCPs) and delayed neutron (DN) monitoring systems were tested continuously to demonstrate the power and utility of the invention. All data used in this investigation were recorded during full-power, steady state operation at EBR-II. The data have been digitized at a 2-per-second sampling rate using 2¹⁴ (16,384) observations for each signal of interest.

FIGS. 1-3 illustrate data associated with the preferred spectral filtering approach as applied to the EBR-II primary pump power signal, which measures the power (in kW) needed to operate the pump. The basic procedure of FIG. 8 was then followed in the analysis. FIG. 1 shows 136 minutes of the original signal as it was digitized at the 2-Hz sampling rate. FIG. 2 shows a Fourier composite constructed from the eight most prominent harmonics identified in the original signal. The residual function, obtained by subtracting the Fourier composite curve from the raw data, is shown in FIG. 3. Periodograms of the raw signal and the residual function have been computed and are plotted in FIG. 4. Note the presence of eight depressions in the periodogram of the residual function in FIG. 4B, corresponding to the most prominent periodicities in the original, unfϊltered data. Histograms computed from the raw signal and the residual function are plotted in FIG. 5. For each histogram shown we have superimposed a Gaussian curve (solid line) computed from a purely Gaussian distribution having the same mean and variance. Comparison of FIG. 5 A and 5B provide a clear demonstration of the effectiveness of the spectral filtering in reducing asymmetry in the histogram. Quantitatively, this decreased asymmetry is reflected in a decrease in the skewness (or third moment of the noise) from 0.15 (raw signal) to 0.10 (residual function).

It should be noted here that selective spectral filtering, which we have designed to reduce the consequences of serial correlation in our sequential testing scheme, does not require that the degree of nonnormality in the data will also be reduced. For many of the signals we have investigated at EBR-II, the reduction in serial correlation is, however, accompanied by a reduction in the absolute value of the skewness for the residual function.

To quantitatively evaluate the improvement in whiteness effected by the spectral filtering method, we employ the conventional Fisher Kappa white noise test. For each time series we compute the Fisher Kappa statistic from the defining equation

k = l where l(ω_k) is the PSD function (see Eq. 14) at discrete frequencies ω_k, and 1(L) signifies the largest PSD ordinate identified in the stationary time series.

The Kappa statistic is the ratio of the largest PSD ordinate for the signal to the average ordinate for a PSD computed from a signal contaminated with pure white noise. For EBR-II the power signal for the pump used in the present example has a K of 1940 and 68.7 for the raw signal and the residual function, respectively. Thus, we can say that the spectral filtering procedure has reduced the degree of nonwhiteness in the signal by a factor of 28. Strictly speaking, the residual function is still not a pure white noise process. The 95% critical value for Kappa for a time series with 2¹⁴ observations is 12.6. This means that only for computed Kappa statistics lower than 12.6 could we accept the null hypothesis that the signal is contaminated by pure white noise. The fact that our residual function is not purely white is reasonable on a physical basis because the complex interplay of mechanisms that influence the stochastic components of a physical process would not be expected to have a purely white correlation structure. The important point, however, is that the reduction in nonwhiteness effected by the spectral filtering procedure using only the highest eight harmonics in the raw signal has been found to preserve the pre-specified false alarm and missed alarm probabilities in the SPRT sequential testing procedure (see below). Table I summarizes the computed Fisher Kappa statistics for thirteen EBR-II plant signals that are used in the subject surveillance systems. In every case the table shows a substantial improvement in signal whiteness.

The complete SPRT technique integrates the spectral decomposition and filtering process steps described hereinbefore with the known SPRT binary hypothesis procedure. The process can be illustratively demonstrated by application of the SPRT technique to two redundant delayed neutron detectors (designated DND A and DND B) whose signals were archived during long-term normal (i.e., undegraded) operation with a steady DN source in EBR-II. For demonstration purposes a SPRT was designed with a false alarm rate, α, of 0.01. Although this value is higher than we would designate for a production surveillance system, it gives a reasonable frequency of false alarms so that asymptotic values of α can be obtained with only tens of thousands of discrete observations. According to the theory of the SPRT technique, it can be easily proved that for pure white noise (such as Gaussian), independently distributed processes, α provides an upper bound to the probability (per observation interval) of obtaining a false alarm—i.e., obtaining a "data disturbance" annunciation when, in fact, the signals under surveillance are undegraded.

FIGS. 6 and 7 illustrate sequences of SPRT results for raw DND signals and for spectrally- whitened DND signals, respectively. In FIGS. 6A and 6B, and 7 A and 7B, respectively, are shown the DN signals from detectors DND- A and DND-B. The steady-state values of the signals have been normalized to zero. TABLE I

Effectiveness of S ectral Filterin for Measured Plant Signals

Normalization to adjust for differences in calibration factor or viewing geometry for redundant sensors does not affect the operability of the SPRT. FIGS. 6C and 7C in each figure show pointwise differences of signals DND-A and DND-B. It is this difference function that is input to the SPRT technique. Output from the SPRT method is shown for a 250-second segment in FIGS. 6D and 7D.

Interpretation of the SPRT output in FIGS. 6D and 7D is as follows: When the SPRT index reaches a lower threshold, A, one can conclude with a 99% confidence factor that there is no degradation in the sensors. For this demonstration A is equal to 4.60, which corresponds to false-alarm and missed-alarm probabilities of 0.01. As FIGS. 6D and 7D illustrate, each time the SPRT output data reaches A, it is reset to zero and the surveillance continues.

If the SPRT index drifts in the positive direction and exceeds a positive threshold, B, of +4.60, then it can be concluded with a 99% confidence factor that there is degradation in at least one of the sensors. Any triggers of the positive threshold are signified with diamond symbols in FIGS. 6D and 7D. In this case, since we can certify that the sensors were functioning properly during the time period our signals were being archived, any triggers of the positive threshold are false alarms.

If we extend sufficiently the surveillance experiment illustrated in FIG. 6D, we can get an asymptotic estimate of the false alarm probability α. We have performed this exercise using 1000-observation windows, tracking the frequency of false alarm trips in each window, then repeating the procedure for a total of sixteen independent windows to get an estimate of the variance on this procedure for evaluating the false alarm probability. The resulting false-alarm frequency for the raw, unfiltered, signals is = 0.07330 with a variance of 0.000075. The very small variance shows that there would be only a negligible improvement in our estimate by extending the experiment to longer data streams. This value of α is significantly higher than the design value of α = 0.01, and illustrates the danger of blindly applying a SPRT test technique to signals that may be contaminated by excessive serial correlation.

The data output shown in FIG. 7D employs the complete SPRT technique shown schematically in FIG. 8. When we repeat the foregoing exercise using 16 independent 1000-observation windows, we obtain an asymptotic cumulative false-alarm frequency of 0.009142 with a variance of 0.000036. This is less than (i.e., more conservative than) the design value of = .01, as desired.

It will be recalled from the description hereinbefore regarding one preferred embodiment, we have used the eight most prominent harmonics in the spectral filtration stage of the SPRT technique. By repeating the foregoing empirical procedure for evaluating the asymptotic values of α, we have found that eight modes are sufficient for the input variables shown in Table I. Furthermore, by simulating subtle degradation in individual signals, we have found that the presence of serial correlation in raw signals gives rise to excessive missed-alarm probabilities as well. In this case spectral whitening is equally effective in ensuring that pre-specified missed-alarm probabilities are not exceeded using the SPRT technique.

In a different form of the invention, it is not necessary to have real sensors paired off to form a difference function. Each single sensor can provide a real signal characteristic of an ongoing process and a record artificial signal can be generated to allow formation of a difference function. Techniques such as an auto regressive moving average (ARMA) methodology can be used to provide the appropriate signal, such as a DC level signal, a cyclic signal or other predictable signal. Such an ARMA method is a well-known procedure for generating artificial signal values, and this method can even be used to learn the particular cyclic nature of a process being monitored enabling construction of the artificial signal.

The two signals, one a real sensor signal and the other an artificial signal, can thus be used in the same manner as described hereinbefore for two (paired) real sensor signals. The difference function Y is then formed, transformations performed and a residual function is determined which is free of serially correlated noise.

Fourier techniques are very effective in achieving a whitened signal for analysis, but there are other means to achieve substantially the same results using a different analytical methodology. For example, filtration of serial correlation can be accomplished by using the ARMA method. This ARMA technique estimates the specific correlation structure existing between sensor points of an industrial process and utilizes this correlation estimate to effectively filter the data sample being evaluated.

A technique has therefore been devised which integrates frequency-domain filtering with sequential testing methodology to provide a solution to a problem that is endemic to industrial signal surveillance. The subject invention particularly allows sensing slow degradation that evolves over a long time period (gradual decalibration bias in a sensor, appearance of a new radiation source in the presence of a noisy background signal, wear out or buildup of a radial rub in rotating machinery, etc.). The system thus can alert the operator of the incipience or onset of the disturbance long before it would be apparent to visual inspection of strip chart or CRT signal traces, and well before conventional threshold limit checks would be tripped. This permits the operator to terminate, modify or avoid events that might otherwise challenge technical specification guidelines or availability goals. Thus, in many cases the operator can schedule corrective actions (sensor replacement or recalibration; component adjustment, alignment, or rebalancing; etc.) to be performed during a scheduled system outage.

Another important feature of the technique which distinguishes it from conventional methods is the built-in quantitative false-alarm and missed-alarm probabilities. This is quite important in the context of high-risk industrial processes and applications. The invention makes it possible to apply formal reliability analysis methods to an overall system comprising a network of interacting SPRT modules that are simultaneously monitoring a variety of plan variables. This amenability to formal reliability analysis methodology will, for example, greatly enhance the process of granting approval for nuclear-plant applications of the invention, a system that can potentially save a utility millions of dollars per year per reactor.

In another form of the invention, an artificial-intelligence based expert system 100 (see FIG. 12) has been developed for automatically configuring a set of sensors A, B, C and D to perform signal validation and sensor-operability surveillance in industrial applications that require high reliability, high sensitivity annunciation of degraded sensors, discrepant signals, or the onset of process anomalies. This expert system 100 comprises an interconnected network of high sensitivity pattern-recognition modules 102 (see FIGS. 9-11). The modules 102 embody the SPRT methodology described hereinbefore for automated parameter surveillance. The SPRT method examines the noise characteristics of signals from identical pairs of sensors 104 deployed for redundant readings of continuous physical processes from a particular industrial device 106. The comparative analysis of the noise characteristics of a pair of signals, as opposed to their mean values, permits an early identification of a disturbance prior to significant (grossly observable) changes in the operating state of the process. As described in more detail hereinbefore, the SPRT method provides a superior surveillance tool because it is sensitive not only to disturbances in signal mean, but also to very subtle changes in the skewness, bias, or variance of the stochastic noise patterns associated with monitored signals. The use of two or more identical ones of the sensors 104 also permits the validation of these sensors 104, j__, determines if the indicated disturbance is due to a change in the physical process or to a fault in either of the sensors 104.

For sudden, gross failures of one of the sensors 104 or components of the system 100, the SPRT module 102 would annunciate the disturbance as fast as a conventional threshold limit check. However, for slow degradation that evolves over a long time period (gradual decalibration bias in a sensor, wearout or buildup of a radial rub in rotating machinery, etc.), the SPRT module 102 provides the earliest possible annunciation of the onset of anomalous patterns in physical process variables. The SPRT-based expert system 100 can alert the operator to the incipience of the disturbance long before it would be apparent to visual inspection of strip chart or CRT signal traces, and well before conventional threshold limit checks would be tripped. This permits the operator to terminate or avoid events that might otherwise challenge technical specification guidelines or availability goals, and in many cases, to schedule corrective actions (sensor replacement or recalibration; component adjustment, alignment, or rebalancing; etc.) to be performed during a scheduled system outage.

The expert system 100 embodies the logic rules that convey to the operator the status of the sensors 104 and the industrial devices 106 connected to a SPRT network 108 (see FIG. 12). When one or more of the SPRT modules 102 indicate a disturbance in sensor signals (i.e., the SPRT modules 102 trip) the expert system 100 determines which of the sensors 104 and/or the devices 106 are affected. The expert system 100 is designed to work with any network of the SPRT modules 102, encompassing any number of the industrial devices 106, process variables, and redundant ones of the sensors 104.

In a most preferred embodiment, the expert system 100 is operated using computer software written in the well known "LISP" language (see Appendix B attached hereto). In this embodiment, the expert system 100 is divided into two segments which act as pre- and post-processors for the SPRT module computer code. The pre-processor section is used to set up operation of the SPRT network 108 and forge connections between the sensor data stream and the SPRT modules 102. The post-processor section contains the logic rules which interpret the output of the SPRT modules 102.

The logic for the expert system 100 depends upon the grouping of the SPRT modules 102. Each of the SPRT modules 102 monitors two identical sensors 104 which measure a physical process variable. (See FIGS. 9-11.) The two sensors can either be redundant sensors 104 on one of the industrial devices 106 or separate sensors 104 on two identical ones of the industrial devices 106 that are operated in parallel. A group of the modules 102 entails all of the connections between the identical sensors 104 for a given physical variable on a group of the industrial devices 106. The number of the modules 102 in a group depends upon the number of identical devices 106 that are operated in parallel and the number of redundant sensors 104 on each device 106 which observe the response of a physical variable. For instance, suppose the expert system 100 is to be applied to an industrial system which contains three identical coolant pumps (not shown). Furthermore, suppose each coolant pump contains two redundant pressure transducers and one thermocouple (not shown). This system 100 would be modeled by two groups of the modules 102. The first group of the modules 102 would connect the six total pressure transducers which measure pump pressure. The second group of the modules 102 would connect the three total thermocouples which measure coolant temperature. For a given group of related sensors 104, the data from each of the sensors 104 are fed into the modules 102. Since the module 102 performs a comparison test between the two sensors 104, the tripping of both of the modules 102 connected to the sensor 104 (in the absence of other tripped modules 102 in the same group) is a necessary and sufficient condition to conclude that the sensor 104 has failed. Therefore, for a group of related sensors 104, the minimum number of modules 102 needed to enable sensor detection is the same as the number of the sensors 104 in the group. For the example discussed above, the number of the modules 102 in the first group would be six, and the number of the modules 102 in the second group would be three.

In applications involving two or more identical ones of the industrial devices 106 operated in parallel and equipped with identical sensors 104, the module 102 applied to pairs of the sensors 104 monitoring the same physical process on the respective devices 106 will provide sensitive annunciation of any physical disturbance affecting one of the devices 106. If each of the devices 106 has only one of the sensors 104 though, it would not be possible for the expert system 100 to distinguish between device degradation and sensor degradation. In this case, the primary benefit of the method would derive from its very early annunciation of a disturbance. For cases in which each of the industrial devices 106 is equipped with multiple, redundant sensors 104, the modules 102 can be applied to pairs of the sensors 104 on each of the industrial devices 106 for sensor-operability verification. In this case, the expert system 100 not only provides early annunciation of a disturbance, but can also distinguish between device degradation and sensor degradation. Moreover, when the expert system 100 determines that the cause of the discrepant signals is due to a degraded one of the sensors 104, it can identify the specific sensor 104 that has failed.

FIG. 9 illustrates the first stage of the expert system 100 processing for a simple generic application involving a single one of the industrial devices 106 that is equipped with triply-redundant sensors 104 for measurement of two physical variables. The expert system 100 first identifies the minimum unique set of signal pairs that will be needed for the network of interacting modules 102. FIG. 10 illustrates a generic application involving two of the industrial devices 106 that are operated in parallel. For this example, it is also assumed that triply-redundant sensors 104 are available for measuring each of two separate physical variables. Once again, the expert system 100 identifies the pair- wise sensor combinations that it uses in building the conditional branching hierarchy for the module configuration. FIG. 11 illustrates a generic application involving three industrial devices 106 that are operated in parallel. Triply-redundant sensors 104 for measuring one physical variable are assumed. The figure shows the pair-wise sensor combinations identified by the expert system 100 for building the conditional branching hierarchy. These figures also depict the three main branches for the logic rules contained in the expert system 100: a grouping of the modules 102 based on a single one of the industrial devices 106, two identical devices 106 operated in parallel or multiple (three or more) devices 106 operated in parallel. The expert system 100 however is not limited to only one of the three cases at a time. The industrial system 100 modeled can contain any number of independent single devices 106, doubly-redundant devices 106 and multiply-redundant devices 106. Each device group, in turn, may contain any number of redundant sensors 104 and any number of physical variables.

The expert system 100 is implemented using a stand-alone computer program set forth in the previously referenced Appendix B. In operation after the program initialization information has been gathered, the computer program prompts the user for the name of a data file that simulates the real-time behavior of the SPRT network 108 connected to the system 100 including the industrial devices 106. The SPRT data file contains space- delimited data values that represent the status of a corresponding module 102. The module 102 has two states: a 0 (non-tripped) state indicates that the signals from the sensors 104 monitored by the module 102 have not diverged from each other, while a 1 (tripped) state indicates that the signals from the sensors 104 monitored by the module 102 have diverged from each other enough to be detected by the SPRT algorithm. Each line of data in the file represents the status of a group of related modules 102 at a given time. Each line contains a list of 0's and l's that correspond to the state of all the modules 102 in the group. The number of groups in the network 108 depends upon the number of groups of identical devices 106 and the number of process variables monitored on each group of devices 106. If the network 108 contains more than one group of related modules 102, the data file will contain a corresponding number of lines to represent the status of all the modules 102 in the network 108 at a given time. For instance, if a system of the industrial devices 106 is modeled by four SPRT groups, the output file will contain four lines of SPRT data for each timestep in the simulation.

Execution of the program of Appendix B includes two procedures. The first procedure (SPRT Expert) provides the instructions for the control of program execution and corresponds to the pre-processor section in an integrated SPRT expert system/SPRT module code. When executed, the procedure first prompts the user to specify the number of device groups in the application. A device group is a group of identical industrial devices 106 (one or more) that are operated in parallel and are equipped with redundant ones of the sensors 104. A device group can contain one or more physical variables. The program then prompts the user for the following information for each of the device groups:

(i) The name of the device group.

(ii) The number of physical variables in the device group.

(iii) The name of the first physical variable in the device group.

(iv) The number of redundant sensors 104 on each device 106 that measures the first physical variable. If the device group contains more than one physical variable, the program will loop through steps (iii) and (iv) to obtain input data for the remaining physical variables. The number of SPRT groups in the application is equal to the sum of the number of physical variables for each of the device groups. Namely,

¹ N ^v SPRT Groups Σ ^ ^J N ' Physical Variables,

Once the program has collected the required data to set up the system, it prompts the user for the name of the SPRT data file. Execution of the program consists of reading the SPRT status values from the data file and evaluating the status of the devices 106 and sensors 104 in the application, as inferred from the SPRT data. Program execution is controlled by a "do" loop. For each pass through the loop, the program reads the data which model the state of each of the SPRT modules 102 in the network at a given time. The SPRT data are then passed to the Analyze procedure. If any of the SPRT modules 102 in the application has tripped (i.e., SPRT value = 1), the Analyze procedure determines which device(s) 106 and/or sensor(s) 104) are affected and reports their status. Looping continues until the end of the data file is reached, upon which the program terminates.

The Analyze procedure contains the logic rules for the expert system 100. It corresponds to the post-processor section in an integrated SPRT expert system/SPRT module code. It is passed lists of O's and l's that represent the status of the SPRT modules 102 at any given timestep of the SPRT program. The number of lists passed to Analyze equals the number of SPRT groups. For each SPRT group, the procedure converts the SPRT data into a list of tripped SPRT modules 102. From the list of tripped SPRT modules 102, the status of the devices 106, and the sensors 104 modeled by the SPRT group are evaluated. Based on the number of devices 106 and the redundant sensors 104 in a SPRT group, the expert system 100 can determine which of the device(s) 106 and or the sensors(s) 104 have failed. In some cases (e.g., if the number of the tripped modules 102 is one, or if the number of redundant sensors 104 in a group is one), the expert system 100 cannot conclude that the device 106 or the sensor 104 has failed, but can only signal that device or sensor failure is possible. Within Analyze, the logic rules are encapsulated by three procedures: SingleDevice, for a SPRT group applied to a single one of the industrial devices 106, DualDevice, for a SPRT group applied to two parallely-operated industrial devices 106; and MultipleDevice, for a SPRT group applied to a group of three or more parallely-operated industrial devices 106.

The following nonlimiting example is illustrative of implementation of the expert system.

Example

The development of a network of the SPRT modules 102 for a general system 110 of the industrial devices 106 and the action of the corresponding logic rules are revealed by an example calculation. The system 110 of industrial devices 106 and a data file representing the transient behavior of the sensors 104 in the system 110 were created. A diagram of the system 110 is shown in FIG. 14 and contains two groups of the industrial devices 106. A first group 112 (identified as turbine devices) contains three of the identical devices 106. Each turbine is equipped with the sensors 104 to measure the steam temperature and steam pressure physical variables. There are two redundant sensors 104 on each turbine reading the steam temperature, while one of the sensors 104 measures the steam pressure. A second device group 114 consists of two coolant pumps. One physical variable, coolant flowrate, is gauged on each coolant pump by a group of four redundant sensors 104. The corresponding network of SPRT modules 102 for the system 110 is shown. Three groups of the SPRT modules 102 are required; with six of the modules 102 in a first module group for the steam temperature sensors on the turbines, three modules 102 in the second module group for the steam pressure sensors 104 on the turbines, and eight of the modules 102 in the third group of the modules 102 for the coolant flowrate sensors 104 on the coolant pumps.

A complete listing of the output from the test run follows hereinafter. Input entered by the user is identified by bold type. From the PC-Scheme prompt, the program is executed by first loading it into memory, and then calling the main procedure (SPRT_Expert). After displaying a title banner, the code asks the user to specify the number of device groups in the network. [2] (load "SPRTEXPT.S")

OK

[3] (SPRT_Expert)

SPRT Expert System Simulation Program Enter the number of device groups -> 2

For each device group in the network, the program requests that the user supply the name of the devices in the group, the number of identical devices, the number of physical variables in the device group, and the names and numbers of redundant sensors for each physical variable. The input entered for device group #1 is: DEVICE NAME:

Enter the name of device group number 1 -> TURBINE DEVICE NUMBER:

Enter the number of devices in the TURBINE device group -> 3

PHYSICAL VARIABLE NUMBER: Enter the number of physical variables in the TURBINE device group -> 2

PHYSICAL VARIABLE NAME: Enter the name of physical variable number 1 of the TURBINE device group -> STEAM TEMPERATURE SENSOR NUMBER:

Enter the number of redundant sensors for the STEAM TEMPERATURE physical variable in the TURBINE device group -> 2 PHYSICAL VARIABLE NAME: Enter the name of physical variable number 2 of the TURBINE device group -> STEAM PRESSURE SENSOR NUMBER:

Enter the number of redundant sensors for the STEAM PRESSURE physical variable in the TURBINE device group -> 1 The input entered for device group #2 is: DEVICE NAME:

Enter the name of device group number 2 -> COOLANT PUMP DEVICE NUMBER:

Enter the number of devices in the COOLANT PUMP device group -> 2

PHYSICAL VARIABLE NUMBER:

Enter the number of physical variables in the COOLANT PUMP device group -> 1

PHYSICAL VARIABLE NAME: Enter the name of physical variable number 1 of the COOLANT PUMP device group -> COOLANT FLOWRATE SENSOR NUMBER:

Enter the number of redundant sensors for the COOLANT FLOWRATE physical variable in the COOLANT PUMP device group -> 4 Once the input data for each device group have been obtained, the program displays a summary of the SPRT network, by identifying the SPRT groups in the network. The number of SPRT Groups in the simulation is 3. SPRT Group #1 contains 3 TURBINE industrial devices with 2 STEAM TEMPERATURE redundant sensors. SPRT Group #2 contains 3 TURBINE industrial devices with 1 STEAM PRESSURE redundant sensor. SPRT Group #3 contains 2 COOLANT PUMP industrial devices with 4 COOLANT FLOWRATE redundant sensors

The final input item required is the name of the data file containing the status values for the SPRT modules in the network. Enter filename for SPRT data -> TEST.DAT

The analysis of the SPRT data is controlled by a do loop. For each pass through the loop, the program retrieves a line of data for each SPRT group in the network. Each block of data retrieved from the file represents the status of all SPRT modules in the network at a moment in time. The program analyzes each block of data to determine whether the SPRT status values imply device and/or sensor failures.

In the test calculation, the first block of data retrieved from the TEST.DAT file is: 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 Analyzing this data, the program reports that:

Analyzing SPRT data set number 1

SPRT Group #1

No SPRTs have tripped

SPRT Group #2

No SPRTs have tripped

SPRT Group #3

No SPRTs have tripped The second block of data retrieved from the file contains some tripped SPRT modules: 100000 000

01000000

Since only one module has tripped in SPRT groups #1 and #3, the program can conclude only that some device and/or sensor failures may have occurred. The program identifies which modules have tripped and which devices or sensors are affected.

Analyzing SPRT data set number 2 SPRT Group #1:

For the STEAM TEMPERATURE physical variable of the TURBINE devices: These 1 of the 6 SPRTs have tripped -> Al-Bl

* * * DEVICE NUMBER A OR DEVICE NUMBER B OF THE TURBINE DEVICES,

SENSOR NUMBER Al, OR SENSOR NUMBER Bl MAY BE FAILING *** SPRT Group #2: No SPRTs have tripped. SPRT Group #3:

For the COOLANT FLOWRATE physical variable of the COOLANT PUMP devices: One SPRT has tripped -> A1-B2

* * * ONE OR BOTH OF THE COOLANT PUMP DEVICES,

SENSOR NUMBER Al OR SENSOR NUMBER B2 MAY BE FAILING *** In the third block of data, additional modules have tripped. 100100 001 01100000 In SPRT group #1, two modules have tripped. Since both of the sensors on device A and both of the sensors on device B are affected, the code concludes that one of the two devices has failed. It cannot identify which of the devices has failed at this time though. In SPRT group #2, one module has tripped. The code concludes that one of the sensors or devices may be failing. Since both modules connected to sensor B2 in SPRT group #3 have tripped, the code concludes that sensor B3 has failed.

Analyzing SPRT data set number 3 SPRT Group #1 :

For the STEAM TEMPERATURE physical variable of the TURBINE devices: These 2 of the 6 SPRTs have tripped - Al-Bl A2-B2

* * * DEVICE NUMBER A OR DEVICE NUMBER B OF THE TURBINE

DEVICES HAS FAILED *** SPRT Group #2:

For the STEAM PRESSURE physical variable of the TURBINE devices: These 1 of the 3 SPRTs have tripped - Cl-Al

* * * DEVICE NUMBER C OR DEVICE NUMBER A OF THE TURBINE DEVICES,

SENSOR NUMBER Cl, OR SENSOR NUMBER Al MAY BE FAILING *** SPRT Group #3:

For the COOLANT FLOWRATE physical variable of the COOLANT PUMP devices: These 2 of the 8 SPRTs have tripped - A1-B2 A2-B2 *** SENSOR NUMBER B2 HAS FAILED *** More modules have tripped in the fourth block of data. 1 0 1 1 0 0 0 1 1

0 1 1 0 0 1 0 0

Since three of the four modules connected to the sensors in device A of SPRT group #1 have tripped, the code concludes that device A has failed. In SPRT group #2, two of the three modules have tripped. But since there is only one sensor per device in this group, the code can only conclude that either a device or sensor failure has occurred. In SPRT group #3, three modules have tripped. The code concludes that one of the two devices has failed.

Analyzing SPRT data set number 4 SPRT Group #1 :

For the STEAM TEMPERATURE physical variable of the TURBINE devices: These 3 of the 6 SPRTs have tripped -> Al-Bl Cl-Al A2-B2

* * * DEVICE NUMBER A OF THE TURBINE DEVICES HAS FAILED * * * SPRT Group #2:

For the STEAM PRESSURE physical variable of the TURBINE devices: These 2 of the 3 SPRTs have tripped -> Bl-Cl Cl-Al

* * * DEVICE NUMBER C OF THE TURBINE DEVICES,

OR SENSOR NUMBER Cl HAS FAILED *** SPRT Group #3:

For the COOLANT FLOWRATE physical variable of the COOLANT PUMP devices: These 3 of the 8 SPRTs have tripped -> A1-B2 A2-B2 A3-B4

* * * ONE OR BOTH OF THE COOLANT PUMP DEVICES HAVE FAILED * * * Notice that two of the tripped modules in SPRT group #3 implicate a failure of sensor B2, the conclusion reached by the analysis of the third block of data. But since an additional module has tripped, the code changes its conclusion from a sensor failure to a failure of one or both of the devices. Although the third module trip may be a spurious trip (i.e., the SPRT modules have a finite false alarm probability) which would mean that the earlier conclusion still holds, the code conservatively concludes that none of the trips are spurious and decides that a device failure has occurred. The code assumes that no module trip is spurious, which causes the code to consistently pick the most conservative conclusion when more than one conclusion can be deduced from the data.

The fifth and last set of data in the file contains additional module trips. 1 1 1 1 0 0 1 1 1

0 1 1 0 0 1 1 0

The additional trips in SPRT group #1 cause the code to conclude that more than one device in the group is affected. In SPRT group #2, all three modules in the group have tripped. Whenever all SPRT modules in a group trip, the code concludes that all devices in the group have failed. In SPRT group #3 the additional module trip does not change the conclusion, since the worst-case conclusion (i.e., one or both of the devices in the group have failed) for the group has already been reached. Analyzing SPRT data set number 5 SPRT Group #1 :

For the STEAM TEMPERATURE physical variable of the TURBINE devices: These 4 of the 6 SPRTs have tripped -> Al-Bl Bl-Cl Cl-Al A2-B2 *** DEVICE NUMBER B OF THE TURBINE DEVICES HAS FAILED ***

* * * SENSOR NUMBER C 1 HAS FAILED * * *

* * * DEVICE NUMBER A OF THE TURBINE DEVICES HAS FAILED * * * SPRT Group #2:

For the STEAM PRESSURE physical variable of the TURBINE devices: All 3 SPRTs have tripped

* * * ALL 3 OF THE TURBINE DEVICES HAVE FAILED * * * SPRT Group #3:

For the COOLANT FLOWRATE physical variable of the COOLANT PUMP devices: These 4 of the 8 SPRTs have tripped -> A1-B2 A2-B2 A3-B4 A4-B4

*** ONE OR BOTH OF THE COOLANT PUMP DEVICES HAVE FAILED *** Notice that the SPRT group #1, the code concludes that devices A and B have failed, but for device C it concludes that only one of its sensors has failed. For device groups containing multiple devices (i.e., three or more identical devices), the code applies its logic rules to each of the devices independently. Since for devices A and B both of their sensors are involved, the code concludes that the devices have failed. For device C only the first sensor is involved, thus the code concludes that only the first sensor on the device has failed. The code reaches the end of the file after analyzing the fifth block of data, causing the code to terminate. For SPRT group #3, a failure of one or both of the devices is indicated, although the pattern of tripped modules can also be interpreted as a simultaneous failure of sensors B2 and B4. The code indicates a device failure because concurrent failures of two or more sensors in a group is deemed to be highly improbable.

While preferred embodiments of the invention have been shown and described, it will be clear to those skilled in the art that various changes and modifications can be made without departing from the invention in its broader aspects as set forth in the claims provided hereinafter. SOFTWARE APPENDIX A

#ιnclude <math . h> ttmclude <stdιo . h> main ()

FILE *falarm, *fopen () ; int fclose () , fprintf () ; static double thresh =■ 4.6, double sigl, sιg2, sιg3 , sιg4 , sιg5 , sig6 ; double dιffl,dιff2,dιff3,dιff4,dιff5,dιff6,dιff7, double spr tl, spr t2 , sp t3 , sp t4 , sp t5 , spr t6 , spr t7 , dumspr t double pi , p2 , 3 , p4 , p5 , p6 , p7 , pO ; int totalm » 0; int num - 0; int aim - 0; double vl, v2, v3 , v4 , v5 , v6 ; double ml , m2 , m3 , m4 , m5 , m6 ; double sfm, gl, g2, g3 , g4 , g5,g6 ,g7 ; long seedl = 13 long seed2 = 23 , long seed3 = 27 long seed4 = 31 long seed5 = 53 long seedδ = 19. extern void normsirnO; extern void sprtO;

* Get the parameters needed in the program */ printf ("\nlnput the sensor failure magnitude SFM."); scanf ("%lf",&sfm) ; printf ("\nlnput the variances of signals 1..6 separated by commas.") scanf ("%lf ,%lf ,%lf ,%lf ,%lf ,%lf " , &vl,&v2, &v3 , &v4 , &v5 , &v6 ) ; printf ("\nlnput the mean of signals 1..6 separated bv commas."); scanf ("%lf,%lf,%lf,%lf,%lf,%lf", &ml , &m2 , &m3 , &m4 , &m5 , &m6 ) ; falarm = fopen ( "alarms .dat" , "w" ; printf ( le. ') fprintf on 1"), fprintf "⁾ fprintf ( falarm, , sfm) fprintf ( falarm, %lf", ml,vl) fprintf ( falarm, %lf", m2, v2) fprintf ( falarm, %lf ", m3,v3) fprintf ( falarm, %lf", m4 , v4) fprintf ( falarm, %lf", m5,v5) fprintf ( falarm, %lf", m6 , v6 ) fprintf

( falarm, printf ("\nStartmg sprt "); pi = 0.0; p2 = 0.0; p3 = 0.0; p4 = 0.0; p5 = 0.0; p6 0 . 0 ; p7 = 0 . 0 ; pO = .0; gi - sf / (vl+v2) ;g2 sfm/ (v3+v4) ;g3 sfm/ (v5+v6 ) g4 = sfm/ (vl+v3) ;g5 sfm/ (v2+v4) ;g6 sfm/ (v3+v5) ; g7 = s fm/ (v4 +v6 ) hile (num < 500)

{ num + = spr tl spr t2 spr t3 spr t4 spr t5 sprtδ = 0.0; sprt7 = 0.0; dumspr t = 0.0; aim = 0 ; do { normsim(&sigl, &sig2 , vl , 2 ,ml,m2 , &seedl, &seed2) no msim (&sig3 , &sig4 , v3 , 4 , m3 , 4 , &seed3 , &seed4) normsim (&sig5 , &sig6 , v5 , v6 , m5 , m6 , &seed5 , &seed6 ) diffl = sigl - sig2; diff2 = sig3 - sig4; diff3 = sig5 - sig6; sprt (sfm, gl, thresh, diffl, &spr tl, &dumspr t) ; spr t (sfm, g2 , thresh, diff2, &spr t2 , &dumspr t) ; sprt (sfm, g3 , thresh, diff3 , &sprt3 , idumsprt) ; } while ((sprtl < thresh) && (sprt2 < thresh) && (sprt3 < thresh)); if ((sprtl =;= thresh) I I (sprt2 =^« thresh) I I (sprt3 =- thresh)) { printf ("\nsprtl=%lf ,sp t2=%lf ,spr 3=%lf " , spr tl, spr t2, sprt3) ; if (sprtl == thresh) do { if ((sprt2 < thresh) && (sprt2 > -thresh)) { normsim (&sig3 , &sig4 , v3 , v4 , m3 , m4 , &seed3 , &seed4) ; diff2 = sig3 - sig4 ; spr t (sfm, g2 , thresh, diff2 , &spr 2 , άdumspr ) ; } if ((sprt3 < thresh) && (sprt3 > -thresh)) { normsim (&sig5 , &sig6 , v5 , v6 , m5 , m6 , &seed5 , &seed6 ) ; diff3 = sig5 - sig6; sprt (sfm, g3 , thresh, diff3 , &spr 3 , &dumspr t) ; } } while ( ( (sprt2<thresh) && (sp t2> - thresh) ) II

( (sprt3<thresh) && (spr t3> - thresh) )) ; else if (sprt2 »- thresh) do { if ( (sprtKthresh) && (spr tl> - th esh) ) { normsim (&sigl, &sig2 , vl , v2 , ml, m2, &seedl, &seed2) ; diffl = sigl - sig2; sprt (sfm, gl, thresh, di fl, &spr l , idumspr t) ; } if ( (sprt3<thresh) && (spr t3> - thresh) ) { normsim (&sig5 , &sig6 , v5 , v6 , m5 , m6 , &seedS , &seed6 ) ; diff3 = sig5 - sig6; sprt (sfm, g3 , thresh, diff3 , &spr t3 , άdumspr t) ; } } while (((sprtKthresh) && (spr tl> - thresh) ) II

( (sprt3<thresh) && (spr 3> - thresh) )) _; else if (sprt3 =- thresh) do { if ( (sprt2<thresh) && (spr t2> - thresh) ) { normsim (&sig3 , &sig4 , v3 , 4 , m3 , m4 , &seed3 , &seed4) ; diff2 = sig3 - sig4; sprt (sfm, g2, thresh, diff2, &sp 2, &dumsprt) ; } if ((sprtKthresh) && (spr tl> - thresh) ) { normsim (&sigl, &sig2, vl , v2, ml , m2, &seedl , &seed2) ; diffl = sigl - sig2; sprt (sfm, gl, thresh, diff1, &sprtl, &dumspr ) ; } } while (( (sprt2<thresh) && (spr t2> - thresh) ) I I

( (sp Kthresh) &&(sprtl> - thresh) ) ) ; printf ("\nsprtl = %lf, sprt2 =%lf, sprt3 = %lf", sprtl, sprt2, sprt3); if ((sprtl =^» thresh) && (sprt2 == -thresh) && (sprt3 = = -thresh)) { do { normsim (&sigl , &sig3 , vl , 3 , ml , m3 , &seedl , &seed3 ) ; diff4 = sigl - sig3; sprt (sfm, g4, thresh, diff4, &spr 4, άdumsp ) ; } while((sprt4 < thresh) && (sprt4 > -thresh)); if (sprt4 < thresh) { do { normsim (&sig2 , &sig4 , v2 , v4 , m2 , m4 , &seed2 , &seed4 ) ; diff5 = sig2 - sig4; sp t (sfm, g5 , hresh, dif f5 , &sp t5 , &dumspr ) ; } while ((sprt5 < thresh) && (sprtδ > -thresh)); if (sprt5 == thresh) aim - 2; } else if (sprt4 == thresh) aim = 1; printf ("\nsp t4 =%lf ,sprt5 =%lf ", sprt4,sprt5) ; printf ("\nAlarm = %d",alm); } else if ((sprt2 == thresh) && (sprtl == -thresh) && (sprt3 == -thresh)) { do { normsim (&sigl , sig3 , vl , v3 , ml , m3 , &seedl , &seed3 ) ; diff4 = sig3 - sigl; sp t (sfm,g4, thresh, diff4 , &spr 4 , &dumsprt) ; } while ((sprt4 < thresh) && (sprt4 > -thresh)); if (sprt4 < thresh) { do { normsim (&sig2 , &sig4 , v2 , v4 ,m2, m4 , &seed2 , &seed4); diff5 = sig4 - sig2; sprt (sfm, g5, thresh, diff5, &sprt5, &dumspr t) ; } while ((sprt5 < thresh) && (sprtδ >- thresh) ); if (sprtS == thresh) alm-4; } else if (sprt4 -= thresh) aim - 3; printf ("\nsprt4 = %lf, sprt5 = %lf", sprt4, sprt5); printf ("\nAlarm = %d", aim) ; } else if ((sprt3 == thresh) && (sprt2 == -thresh) && (sprtl == -thresh)) { do { normsim (&sig3 , &sig5 , v3 , v5,m3 , m5 , &seed3 , &seed5); diff6 = sig5 - sig3 ; sprt (sfm,g6 , th esh, diff6 , &sprt6, &dumspr ) ; } while((sprt6 < thresh) && (sprtδ >-thresh)); if (sprtδ < thresh) { do

{ normsim(&sig4 , &sig6,v4,v6 m4 , m6 , &seed4 , &seed6 diff7 = sig6 - sig4; sp t (sfm, g7, thresh, diff , &sp t7 &dumspr t) ; } while((sprt7 < thresh) && (sprt7 >-thresh)) if (sprt7 == thresh) aim = 6; else if (sprt6 == thresh) aim = 5 ; printf ("\nsp t6 - %lf, sprt7 = %lf", sprtδ, sprt7) printf ("\nAlarm = %d", aim) ; } else aim = 7 if (alm==l) Pi if (alm==2) P2 if (alm==3) P3 if (alm==4) P4 if (alm»=5) P5 if (alm==6) P6 1.0 if (alm==7) 7 1.0 totalm

}

Pi pl/totalm P2 p2/totalm P3 p3/totalm p4 p4/totalm P5 p5/totalm p6 p6/totalm p7 p7/totalm fprintf (falarm, "\Empirical probabilities : ") ; fprintf (falarm, "\n Total number of diagnostic run %d", totalm) fprintf (falarm, "\n P{Alarm 1 sounded} = %lf",pl) fprintf (falarm, "\n P{Alarm 2 sounded} = %lf",p2) fprintf (falarm, "\n P{Alarm 3 sounded} - %lf",p3) fprintf (falarm, "\n P {Alarm 4 sounded} = %lf",p4) fprintf (falarm, "\n P{Alarm 5 sounded} = %lf",p5) fprintf (falarm, "\n P{Alarm 6 sounded} = %lf",p6) fprintf (falarm, "\n P{Alarm 7 sounded} = %lf",p7) fclose (falarm)

In running this code for the nuclear plant data, two things need Ko be changed when changing days: the filename and the title. The filename to be changed is at the top of the following code on the first line of the code (DO NOT CHANGE THE DIRECTORY) . The title to be changed is on the 6th line of code. To use the mean for the current day, leave all procedures in as they are. To use the mean of the day ran previously, comment out the 7-9th lines o f code. This will cause the means of the data to be read from a file created at the last run - this way you can compare the whole month by using the mean of the first day. Output is directed to the file ' spr tfp.1st' . In this file, the results of the sprt run including longest sequence of failures and total number of decisions for each of the signal pairs is output. The signal pairs are cl-c2, c3-c4, c5 -c6, c7-c8, c9-cll, and cl0-cl2. In addition to the cumulative resul ts of the run, the SPRT1 and SPRT2 values at each data point are also given (to remove this feature, comment out the 'proc print' procedure on the last line of the macro) .

Written by Kristin K. Hoyer

filename sprtfil ' "hoyer/data/fp/spr vals .1st ' ; libname datlib ' "hoye /data/fp' ; data signal; proc means mean noprint; var cl c2 c3 c4 c5 cδ c7 c8 c9 clO ell cl2 cl3 cl4 cl5 cl6 cl7 cl8 cl9 c20 c21 c22; output out=datlib. fpmeans mean=ml m2 m3 m4 m5 m6 m7 m8 m9 mlO mil ml2 ml3 ml4 ml5 ml6 ml7 ml8 ml9 m20 m21 m22; data signal; set signal; i - 1; set datlib. fpmeans poιnt=ι; diffl - cl-ml- (c2-m2) ; diff2 c2-m2 - (c3 -m3) ; diff3 c4 -m4 - (c5 -m5) ; diff4 c5 -m5 - (c6 -m6) ; diffδ c7 -m7 - (c8-m8) ; diffδ c9-m9- (clO-mlO) ; diff7 cll-mll- (cl2-ml2) diffβ cl3-ml3- (cl4-ml4) diff9 cl5-ml5- (clδ-ml6) . difflO = cl7-ml7 - (cl8-ml8) diffll - cl9-ml9- (c20-m20) diffl2 = c21-m21- (c22-m22) keep diffl diff2 dιff3 diff4 dιff5 diffβ dιff7 diffδ dιff9 difflO diffll diffl2; proc means data=sιgnal var noprint; var diffl dιff2 dιff3 dιff4 dιff5 diffδ dιff7 dιff8 dιff9 difflO diffll diffl2; output out=datlib. fpvars var=vl v2 v3 v4 v5 vδ v7 v8 v9 vio vll vl2; data signal; set signal; i-1; set datlib. fpvars poιnt=ι; sfml = 4.5*vl**0 5 sfm2 = 2.5*v2**0 5 sfm3 = 3.5*v3**0 5 sfm4 = 2.5*v4**0 5 sfm5 = 2.5*v5**0 5 sfmδ = 2.5*v6**0 5 sfm7 = 2.5*v7**0 5 sfmθ = 2.5*v8**0 5 sfm9 = 2.5*v9**0 5 sfmlO = 2.5*vl0**0 5; sfmll = 2.5*vll**0 5; sfml2 - 2.5*vl2**0 5; output

*_****_************_{********************}*_***_{* * * *}

* This is the updated, working SPRT macro. _{* * * * * * *} * . In order to use this macro, several

* things need to be set up in the main program,

* Call a data step and retrieve the data, the

* sensor failure magitude, and the variance of

* the signals The variable must be renamed

* as follows: y - data, sfm=sensor failure magn . _{* * *}

* v^vaπance . To change the threshhold limi

* for the SPRT change the value of the macro _{* * *}

* variable thr esh (m the SPRT macro) to the

* desired valu _{* * *}•

macro do2sprt g^sfm/v; thrsh - 6.9; wndw - 50000; if (_N_ = 1) then do; retain sprtl 0;retain sprt2 0 ; retain faill 0;retain fail2 0; retain faillpt 0; retain fail2pt 0; retain fseqpt 0; retain failseq 0; retain fl 0; retain f2 0; retain flpt 0; retain f2pt 0; end; retain ypl; retain yp2 ;

INC1= -G*( (SFM/2) -Y) ; INC2 = - G* ( (SFM/2 ) +Y) ; if (ypl>= thrsh or ypl<= -thrsh) then sprtl = incl; else sprtl = ypl + incl; if (yp2>= thrsh or yp2<= -thrsh) then sprt2 = inc2; else sprt2 = yp2 + inc2; ypl = sprtl ;yp2 = sprt2;

IF SPRTK-- thrsh THEN SPRTK- thrsh ; IF SPRT2 <= -thrsh THEN SPRT2= - hr sh; IF SPRT1>= thrsh THEN SPRTKthrsh; IF SPRT2>= thrsh THEN SPRT2 = thrsh ; IF (SPRT1 >= thrsh) THEN TRIPH1 + 1; IF (SPRT2 >=• thrsh) THEN TRIPH1 + 1; IF (SPRT1 <= -thrsh) THEN TRIPH2 + 1; IF (SPRT2 <= -thrsh) THEN TRIPH2 + 1; IF (SPRT1 <- -thrsh) THEN fl = 0; ELSE IF (SPRT1 >= thrsh) THEN DO; Fl= faill + 1;

IF (Fl = 1) THEN F1PT = _N_; END; ELSE DO;

Fl = faill; F1PT = faillpt; END; IF (SPRT2 <■ -thrsh) THEN f2 - 0; ELSE IF (SPRT2 >- thrsh) THEN DO; F2 = fail2 + 1; IF (F2 = 1) THEN F2PT - _N_; END; ELSE DO;

F2 = fail2; F2pt - fail2pt; END; IF (Fl >= FAILSEO) OR (F2 >= FAILSEQ) THEN DO; IF (Fl >= F2) THEN DO;

FAILSEO - Fl; FSEQPT = F1PT;END; ELSE DO;FAILSEQ = F2;FSEQPT = F2PT;END; END;

ELSE DO; FAILSEQ + 0; FSEQPT + 0;END; faill = fl;fail2 = f2; faillpt = f lpt;fail2pt = f2pt; CNT + 1;

VAL = MOD (_N_, wndw) ; IF VAL = 0 OR EOF THEN DO; FILE PRINT; IF (CNT > 0) THEN DO;

FREQH1 = TRIPH1 / CNT;

FREQH2 = TRIPH2 / CNT;

TOTHl + FREQHl;TOTH2 + FREQH2 ; TOTCNT + 1;

TH1 = TOTHl / TOTCNT; TH2 = TOTH2 / TOTCNT;

ADDTRIP = FREQH1+FREQH2;ADDT=TH1+TH2;

ALPHKTOTH1/ (TOTH1+TOTH2 ) *100.0; BETl = TOTH2 / (TOTHl +TOTH2 ) ;

PUT ' TRIPPING FREQUENCY- WINDOW SIZE: ' CNT/ HI TRIPPING FREQUENCY / OBSERVATION (-> FAILURE) FREQH1/ H2 TRIPPING FREQUENCY / OBSERVATION ( =>NORMAL) : FREQH2/ COMBINED TRIPPING FREQUENCY PER OBSERVATION: ADDTRIP/ LONGEST SEQUENCE OF FAILURE DECISIONS: FAILSEQ/ DATA POINT AT WHICH SEQUENCE FIRST OCCURRRED : FSEQPT/ PERCENT TIME A FAILURE MODE SELECTED: ALPH1// triphl 0;triph2 = 0;cnt faill fail2 = 0; faillpt = 0;fail2pt 0;

END; END; drop cnt totcnt tothl toth2 triphl triph2 failseq fseqpt; drop fail 1 faillpt fail2 fail2pt fl flpt f2 f2pt; proc means mean noprint; var sprtl; output out=dummy ; %

data; set signal; set end=EOF; y=diff3 ; sfm=sfm3 ; v=v3 ; do2spr t

%macro fourrec;

/* SAS CODE FOR THE SPECTRAL DECOMPOSITION/RECONSTRUCTION */

/* This macro constructs a curve from raw data using the */

/* highest &M Fourier modes (calculated from the power spectral */

/* density). It reads raw data from the file 'kris' in the

/* SAS library assigned to the libref 'datlib' */

/* and places the constructed curve in the file 'datlib */

/* recons** 'where ** is the reactor variable code number. */

/* Both files are SAS datasets . The macro also performs a */

/* Fisher Kappa Whitetest on the periodograms of the original */

/* and residual data. */

/* Written by Kristin Hoyer */

/* Using a number of pts that is a power of 2 to speed up the */ /* spectra procedure -- K=16384 */ proc spectra data=datlib.kr is out=tspec coef p s whitetest; var &rawdat; proc sort data=tspec out=so spec; /* Sort by descending psd to get*/ by descending p__01 s_01; /* most prominent periodics */ data datlib. &outset; set datlib. kris; min = _N_ - 1; do i=l to 9; /* Use 8 highest psd vals */ set sortspec point=i; /* Access ith Fourier coef*/ if( i - 1) then powe =cos_01/2.0; /* and freq and reconstr. */ else do; /* the curve. */ power=power + cos_01*cos (freq* min) +sin_01*sin (freq*min) ; end; end; /* Fourier curve = power */ min = min*l.0/120.0; /* residual = noise */ noise = &rawdat - power; keep min power &rawdat noise; output; if _N_>16383 then stop; proc spectra data= datlib. ϋdset out=newpsd whitetest p; var noise; data newpsd; set newpsd; p_res = p_01; keep p_res freq; data datlib. &specset; merge newpsd tspec; by freq; keep freq p_01 p_res; %mend four ec;

%macro addran;

/* This macro adds Gaussian random noise to the Fourier curve *_/'

/* 'power' to create the reconstructed curve 'recons' with the */

/* same mean and variance as the original curve 'rawdat'. */

/* The input dataset should come from the library with */

/* libref = 'datlib', and the dataset name should be assigned */

/* (with a macro) the the reference name 'dset' */ proc means data=datlib. &outset var noprint; /* Get the variance of the */ var noise; /* residual function */ output out=twork var=sιgmasq; data datlib. &outset; l-l; set datlib. &outset; set twork poιnt=ι; /'Generate the random data */ recons = sιgmasq**0.5 *rannor(3) + power; /*and add to 'power */ keep min &rawdat power noise recons; proc means data=datlιb. &dset mean var skewness kurtosis min max; %mend addran;

%macro normts (x=x) ;

/* This macro tests the normality of the data 'x' */

/* passed as a paramete . The methods used are */

/* the D'Agostinσ Pearson K^Λ2 test for the 3rd and 4th */

/* moments and the Kolmogorov-Smirnov test. */

/* The input dataset should come from the library with */

/* libref = 'datlib', and the dataset name should be assigned */

/* (with a macro) the the reference name 'dset' */

/* Written by Kristin Hoyer */ data twork; set datlib. Soutset; proc means data=twork n nmiss skewness kurtosis std noprint; var &x; output out=stats n=n skewness-sk kurtosιs=k std=sιgma;

/* Compute the skew contribution to the test statistic */

/* according to the Johnson Su approximation */ data skew; set stats; y = sk* ( (n+1) * (n+3) / (6.0* (n-2) ) ) **0.5; beta2 - 3* (n*n+27 *n-70) * (n+1) * (n+3 ) / ( (n- 2) * (n+S ) * (n+7 ) * (n+9 ) ) ; wsqr = -1.0 + (2.0*(beta2 - 1.0))**0.5; delta = 1.0/ (0.5*log(wsqr ) ) **0.5; alpha = (2.0/ (wsqr - 1.0))**0.5; z = delta-log(y/alpha + (y*y/ (alpha*alpha) + 1.0)**0.5); xsqrbl = z*z; keep xsqrbl sk; output;

/* Compute the kurtosis contribution to the test statistic */

/* according to the Anscombe and Glynne Approximation */ data kurt; set stats; k = (k+3.0*(n-l)**2/( (n-2)* (n-3) ) ) ; b2bar = 3.0* (n 1) /(n+1) ; varb2 = 24.0*n* (n-2) * (n-3) / ( (n+1) **2 * (n+3) * (n+5) ) , x = (k - b2bar ) /varb2**0.5; beta = 6.0* (n*n-5*n+2) / ( (n+7) * (n+9) ) * (6.0* (n+3) * (n+5)

/(n*(n-2) * (n-3) ) )**0.S; a = 6.0+8.0/beta* (2.0/beta + (1.0+4.0/ beta**2 )**0.5); dum = (1.0-2.0/a) / (1.0+x* (2.0/ (a-4.0) ) **0.5) ; dum3= (abs (dum) ) **0.333333; if (dum < 0.0) then dum3 = -dum3; z = 1.0 - 2.0/(9.0*a) dum3; z = z / (2.0/ (9.0*a) ) **0.5; xsqrb2 = z**2; keep xsqrb2 k; output; /* Get test statistic. It is Chi square with 2 degrees of freedom */ data _null_; file print; merge skew kurt; ksqr = xsqrbl + xsqrb2; alpha - probchi (ksq , 2, 1) ; put 'D Agostmo Pearson results for ' "&x"// 'Skew = ' sk// 'Kurtosιs= ' k// 'Test value ' ksqr// 'Alpha = ' alpha; proc univaπate data=datlιb. &outset normal, var &x; %mend normtst;

%macro runs (r =r ) ;

/* This macro performs a 'runs' test on the variable r to determine */

/* autocor elation. This method was taken from "Checking for */

/* Autocorrelation in Regression Residuals" bv Dickey and Brocklebank*/

/* The input dataset should come from the library with */

/* libref = 'datlib', and the dataset name should be assigned */

/* (with a macro) the the reference name 'dset' */

/* Written by Kristin Hoyer */ data _null ; file print; set end=eof; if &r = . then goto miss; /* Get negative and positive runs */ pr=&r>=0; /* and missing values. */ np+pr ; nn+ (1-pr ) ; if pr ne lag(pr) then runs + 1; miss: if eof then do; put ' Run of Signs Test'//

'Number of Runs ' runs//

'Number of Negative Residuals ' nn//

'Number of Positive Residuals ' np;

/* Calculate the Z Statistic */ u=2*nn*np/ (nn+np) +1 ; s=sqrt (2*nn*np* (2*nn*np-nn-np) / ( (nn+np 1) * (nn+np) **2) ) , zp= (runs- u+ .5) /s;pp=probnorm (zp) ; zn= (runs u- .5) /s ;pn=l -probnorm (zn) ; put 'Test for positive autocorrelation '/

'Z= ' zp 'Prob < Z= ' pp//

'Test for negative autocorrelation'/

'Z= ' zn 'prob > Z= 'pn; end; %mend runs; %include macros; %let dset=reconsl4 ; %let rawdat=cl4; %let specset=psdl4 ; %let &M = 10; %fourrec %addran

%normtst (x = &rawdat) %normtst (x=noise) proc means data=data. &dset mean noprint; var &rawdat; output out=twork mean=mu; /* Get the residuals needed */ data; /* for the 'sign' macro */ set data.&dset; i-1; set work point=i r= &rawdat - mu; res=noise; keep r res; ouput ;

%runs (r =r )

%runs (r =res)

APPENDIX B Listing of the SPRTEXPT.S File

SPRT_Exper Procedure

SPRT_Expert Procedure (define (SPRT_Exper t)

RetrieveData Procedure - This internal procedure retrieves the SPRT data from the specified DataFile. The procedure reads the next NLines lines from the file. The procedure is recursive, it decrements NLines until NLmes reaches 1. If the end-of-file is reached during the data re trieval, the eof character is added to the end of the list returned bv the procedure. The procedure returns the NLines read as a list (define RetrieveData

(lambda (NLines DataFile)

(let ( (CurrentLine (read line DataFile) ) ) Read next line in DataFile (if (eof - object? CurrentLin If eof reached, return eof CurrentLine character . (if (eq? 1 NLines) , If Nlines read, return null

(list (Chars - to-List (string list CurrentLine)))

(cons (Chars - to- List (string list CurrentLine))

(RetrieveData (- Nlines 1) DataFile) ) ) ) ) ) )

Chars - to-List Procedure - This internal procedure recursively converts a list of characters that contain embedded 0 and 1 characters into a list of 0 s and 1 s. Any other character than a 0 or 1 is ignored (define Chars- o List (lambda (Chars)

(cond ((null⁷ Chars) ()) ; If end of list, return null,

((char=? (car Chars) # 0) ; Add 0 to list

(cons 0 (Chars to-List (cdr Chars))) ) ((char-? (car Chars) # 1) ; Add 1 to list.

(cons 1 (Chars to-List (cdr Chars))) ) (else (Chars- to-List (cdr Chars))) ) ) )

Eof-List? Procedure - This internal procedure searches a list element by element for the eof character. It returns true if the end-of file character is found, or false if the list does not contain the eof character. (define Eof-List? (lambda (Lst)

(if (eof -obuect? Lst) If the current object is the eof #t character, then return true.

(if (null? Lst) If the list is null, 0 return false. (Eof List? (cdr Lst) ) ) ) )

ListTrippedSPRTs Procedure This internal procedure processes the list of SPRT data for a pass through the main do loop. Each element of the list contains the SPRT data for an independent group of SPRTs. Each element of the list is processed by the internal FindSPRTs procedure, which returns a list of numbers where each number reprsents the SPRTs that have tripped [i.e., equal to 1]. Thus if the following list of SPRTs is passed to FindSPRTs: [0 1 0 0 0 1 0 1], it would return the ; ; following list: [2 6 8]. (define ListT lppedSPRTs

Main body of ListTr lppedSPRTs

This protion of the procedure passes each element of SPRT_Lιst to FindSPRTs and returns a list of the results from FindSPRTs (lambda (SPRT_Lιst) (if (null? SPRT_Lιst) 0 (cons (FindSPRTs (car SPRT_Lιst) 1)

(ListTr lppedSPRTs (cdr SPRT_Lιst) ) ) ) ) )

FindSPRTs internal procedure: (define FindSPRTs (lambda (Lst I)

(cond ((null? Lst) ()) ; Return null if end of Lst is reached.

( (eq? (car Lst) 1) ; If SPRT is tripped,

(cons I ; add number to output list.

(FindSPRTs (cdr Lst) (+ 1 1)) ) )

(else ; If SPRT has not tripped, ignore

(FindSPRTs (cdr Lst) (+ 1 1)) ) ) ) )

DisplayGroup internal procedure

This procedure displavs the input data for each SPRT group (define DisplayGroup

(lambda (GroupData Count) (if (null' (car GroupData))

0 ; Return null at end of GroupData list

(begin

(newline) (display SPRT Group # ) (display Count) (newlme) (display contains ) (displav (caar GroupData) ) (displav ) (display (caddar GroupData) ) (if (eq? (caar GroupData) 1)

(display industrial device ) (display industrial devices ) ) (newline) (display with ) (displav (cadar GroupData) ) (display ) (display (car (cdddar GroupData) ) ) (if (eq? (cadar GroupData) 1)

(display redundant sensor ) (display redundant sensors. ) ) (newline) (DisplayGroup (cdr GroupData) (+ 1 Count)) ) ) ) )

Main body of the SPRT_Expert Procedure. First, mtialize local variables, (let ( (NSPRT_Groups 0) (NDeviceGr oups 0) (NVariables 0)

(NDevices 0) (NSensors 0) (Count 0)

(DName ) (VName )

(InputFile ) (InputPort )

(GroupData ()) (SPRT_Data ()) )

Print program title,

(newlme) (newlme) (newline)

(display SPRT Expert System Simulation Program )

(newlme) (newline) (newline)

Query user for number of device groups to be analyzed

(display Enter the number of device groups ) (set! NDeviceGroups (string number (read line) e d) )

Start do loop to input data for each device group. (do ( (I 1 (+ I 1)) )

( ( NdeviceGroups I) #t )

Body of do loop, (newl e) (newlme)

(display

(newlme)

;; Query user for device group names.

(display DEVICE NAME: ) (newline)

(display Enter the name of device group number ) (display I)

(display )

(set! DName (read-line))

(newlme)

;; Query user for the number of independent devices in the group.

(display DEVICE NUMBER: ) (newline)

(display Enter the number of devices in the ) (display DName)

(newline) (display device group - )

(set! NDevices (string- number (read-line) e d) )

(newlme)

;; Query user for number of physical variables in device group.

(display PHYSICAL VARIABLE NUMBER: ) (newlme)

(display Enter the number of physical variables in the )

(display DName) (newline) (display device group )

(set¹ NVaπables (string number (read line) e d) )

Start do loop to input data for each device/ phvsicai variable group, (do ( (J 1 (+ J l) ) ) ( ( Nvariables J) #t ) Body of do loop.

(newlme)

Query user for physical variable names.

(display PHYSICAL VARIABLE NAME: ) (newline)

(display Enter the name of physical variable number )

(display J) (newlme)

(display of the ) (displav DName)

(display device group )

(set! VName (read-line))

(newlme)

Query user for the number of reαundant sensors the qroup

(display SENSOR NUMBER. ) (newlme)

(display Enter the number of redundant sensors for the )

(display VName) (newline)

(display physical variable in the ) (display DName)

(displav device group - )

(set! NSensors (string- number (read-line) e d) )

;; Increment the number of SPRT groups, (set! NSPRT_Groups (+ NSPRT_Groups 1))

;; Add SPRT group input data to the GroupData list, (set! GroupData (cons

(list NDevices NSensors DName VName)

GroupData) ) ) )

;; Reverse the order of the GroupData list so that it will be consistent

;; with the SPRT data.

(set! GroupData (reverse GroupData))

Display SPRT network data, (newline) (newline)

(displav +++++++++₊++++++++++++++++++++++++++++++++₊+_{+++++++++++++++ j} (newline) (newlme) (displav The number of SPRT Groups in the simulation is ) (display NSPRT_Groups) (display . ) (newline)

Call DisplayGroup to display the input variables for each SPRT group (DisplayGroup GroupData l) (newline) (newline) ₍display _{+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++} >

(newline) (newline)

; ; Query user for the name of the data file that contains the SPRT data

(display Enter filename for SPRT data - )

(set! InputFile (read-line))

;; Open SPRT data file.

(set! InputPort (open- input- file InputFile))

Enter main do loop, which retrieves all of the SPRT data written at a timestep. For each timestep of the SPRT program, a number of lines of data are written to the data file. Each line contains the SPRT data [0 s and 1 s] for an independent group of SPRTs. The number of lines written during each timestep thus equals NSPRT_Groups . (do ( (InputData (RetrieveData NSPRT_Groups InputPort)

(RetrieveData NSPRT_Groups InputPort)) ) ( (Eof-List? InputData) (newline) (newline)

(display End of ) (display InputFile)

(display reached -- SPRT_Expert terminates. ) (newline) )

Main body of the do loop .

Update count, which counts the sets of SPRT data written to the file.

(set¹ count (+ count 1)) ; ; Convert the SPRT data for this pass through the loop to a list of tripped SPRTs [i.e., SPRT = 1] . (set! SPRT_Data (ListTr lppedSPRTs InputData)) (newlme) (display (newlme) (newline) (display Analyzing SPRT data set number ) (display count) (newline) (newlme)

Call the SPRT data analysis procedure. This procedure analvzes the list of tripped SPRTs. Each element in the list consists of list of tripped SPRTs. There are NSPRT_Groups elements the list - one for each corresponding group of independent SPRTs (Analyze GroupData SPRT_Data 1)

; ; Pause the screen so that the user can read the results for tne

;; current block of data.

(display Hit the enter key to analyze the next )

(display block of data ) (newlme) (read line) (newline) ) ) )

End of SPRT_Expert Procedure

Analyze Procedure

Analyze Procedure This procedure contains the logic rules for the SPRT Expert System Simulation program.

(define (Analyze GroupList SPRT_Lιst GroupCount)

AnalyzeGroup procedure - This internal procedure controls the flow of SPRT data analysis. It is passed the input data and the list of tripped SPRTs for a group of independent SPRTs. Depending upon the number of devices in the group, AnalyzeGroup calls a procedure which performs the actual analysis of the data, (define AnalyzeGroup

(lambda (Inputs SPRTs Count)

(display SPRT Group # ) (displav Count) (display : ) (newline) (cond ( (eq? (car Inputs) 1) ; The number of devices is 1.

(if (null? SPRTs) ; If no SPRTs have tripped,

(begin

(display No SPRTs have tripped. ) (newlme) (newline) () ) ; return null .

(SingleDevice (cdr Inputs) SPRTs) ) ) ( (eq? (car Inputs) 2) ; The number of devices is 2.

(if (null? SPRTs) ; If no SPRTs have tripped,

(begin

(display No SPRTs have tripped. ) (newline) (newline) () ) ; return null.

(DualDevice (cdr Inputs) SPRTs) ) ) (else ; The number of devices = 3.

(if (null? SPRTs) ; If no SPRTs have tripped,

(begin

(display No SPRTs have tripped. ) (newlme) (newline) () ) ; return null.

(MultipleDevice Inputs SPRTs) ) ) ) ) )

SingleDevice procedure - This procedure contains the logic for the analysis of a SPRT group which has 1 device, (define SingleDevice

(lambda (InputData SPRT_Data)

First use temporary variables to extract the data from the InputData list and the number of tripped SPRTs. (let ( (NSensors (car InputData)) (DName (cadr InputData) ) (VName (caddr InputData) ) (NSPRTs (length SPRTJData) ) ) ;; The logic depends upon the number of tripped SPRTs. (cond ( ( = NSPRTs 3)

;; In this case, 3 or more SPRTs have tripped. First write message showing current device and physical variable, (display For the ) (display VName)

(display physical variable of the ) (display DName) (display device: ) (newline)

(if (eq? NSPRTs NSensors) ; If all of the sensors failed, (begin ; write this message,

(display All ) (display NSPRTs) (display SPRTs have tripped ) ) (begin ; or, write this message. (display These ) (display NS PRTs ) (display o f the )

(di splay NSensor s ) (display S PRTs have tr ipped )

(newl ine)

(DisplaySPRTs 1 NSensors SPRT_Data) ) ) We conclude that the device has failed, (newline) (display *** THE ) (display DName) (displav DEVICE HAS FAILED *** ) (newline) (newl e) )

( (eq? NSPRTs 2) ,; In this case, 2 SPRTs have tripped. First write message showing current device and physical variable, (display For the ) (display VName)

(display physical variable of the ) (display DName) (displav device: ) (newlme)

List the tripped SPRTs. (display These 2 of the ) (display NSensors) (display SPRTs have tripped ) (newline) (DisplaySPRTs 1 NSensors SPRT_Data) (newline)

Next, determine if a sensor or the device has failed, (cond ( (eq? (car SPRT_Data) ( (cadr SPRT_Data) 1))

We can conclude that a sensor has failed, because the tripped SPRTs are next to each other (display *** SENSOR NUMBER A )

(display (cadr SPRT_Data) ) (displav HAS FAILED *** (newlme) (newlme) ) ((and (eq' (car SPRT_Data) 1)

(eq? (cadr SPRT_Data) NSensors)) We can conclude that the first sensor has failed, because the first and last SPRTs have tripped, (display *** SENSOR NUMBER Al HAS FAILED *** ) (newline) (newline) ) (else

We can conclude that the device has failed, since the tripped SPRTs are not adjacent (display *** THE ) (display DName) (displav DEVICE HAS FAILED *** ) (newline) (newlme) ) ) )

( (eq? NSPRTs 1)

In this case, 1 SPRT has tripped This is due either tc a early indication of a sensor or device failure, or to a spurious SPRT trip. We conclude a possible failure (display For the ) (display VName)

(display physical variable of the ) (display DName) (display device: ) (newline) (display One SPRT has tripped ) (DisplaySPRTs 1 NSensors SPRT_Data) (newline) (display *** THE ) (display DName) (display DEVICE, ) (if (eq? (car SPRT_Data) NSensors) (begin

(display SENSOR NUMBER Al, ) (newlme) (display OR SENSOR NUMBER A )

(display NSensors) (display MAY BE FAILING *** ) (newline) (newline) ) (begin

(display SENSOR NUMBER A ) (display (car SPRT_Data) )

(display , ) (newline)

(display OR SENSOR NUMBER A )

(display (+ (car SPRT_Data) 1))

(display MAY BE FAILING *** )

(newl e) (newlme) ) ) ) ) ) ) ) DualDevice procedure This procedure contains the logic for the analysis of a SPRT group which has 2 devices (define DualDevice

(lambda (InputData SPRT_Data)

;; First use temporary variables to extract the data from the InputData list and the number of tripped SPRTs. (let ( (NSensors (car InputData)) (DName (cadr InputData) ) (VName (caddr InputData) ) (NSPRTs (length SPRT_Data) ) ) The logic depends upon the number of tripped SPRTs. (cond ( ( = NSPRTs 3)

In this case, 3 or more SPRTs have tripped. First write message showing current device and physical variable (displav For the ) (displav VName)

(display physical variable of the ) (display DName) (display devices: ) (newline)

(if (eq? NSPRTs (* NSensors 2)) ; If all of tne sensors failed, (begin ; write this message,

(display All ) (display NSPRTs) (display SPRTs have tripped ) ) (begin ; or, write this message

(display These ) (display NSPRTs) (display of the ) (display (* NSensors 2))

(displav SPRTs have tripped ) (newline) (DisplaySPRTs 2 NSensors SPRT_Data) ) ) We conclude that the device has failed, (newlme) (display *** ONE OR BOTH OF THE ) (display DName) (display DEVICES HAVE FAILED *** ) (newline) (newline) )

( (eq? NSPRTs 2)

In this case, 2 SPRTs have tripped. First write ;; message showing current device and physical variable, (display For the ) (displav VName)

(display physical variable of the ) (display DName) (displav devices: ) (newline)

List the tripped SPRTs. (display These 2 of the ) (displav (* NSensors 2)) (display SPRTs have tripped ) (newline) (DisplaySPRTs 2 NSensors SPRT_Data) (newline) ;, Next, determine if a sensor or the αevice nas failed (cond ( (eq? (car SPRT_Data) ⁽ (cadr SPRT_Data) 1))

; ; We can conclude that a sensor has failed, because ;; the tripped SPRTs are next to each other, (display *** SENSOR NUMBER ) (if (even? (car SPRT_Data) ) (begin

(display B )

(display (+ (/ (car SPRT_Data) 2) 1)) ) (begin

(display A )

(display (+ (quotient (car SPRT_Data) 2) l) ) ) ) (display HAS FAILED *** ) (newline) (newlme) ) ((and (eq' (car SPRT_Data) 1)

(eq' (cadr SPRT_Data) (* NSensors 2))) We can conclude that the first sensor of the second device has failed, because the first and last SPRTs have tripped, (display *** SENSOR NUMBER Bl HAS FAILED *** ) (newline) (newl e) ) (else

; ; We can conclude that the device has failed, since the tripped SPRTs are not adjacent, (display *** ONE OR BOTH OF THE ) (display DName) (display DEVICES HAVE FAILED (newlme) (newline) ) ) )

( (eq? NSPRTs 1)

In this case, 1 SPRT has tripped. This is due either to a early indication of a sensor or device failure, or to a spurious SPRT trip. We conclude a possible failure (display For the ) (display VName)

(display physical variable of the ) (display DName) (display devices: ) (newlme) (display One SPRT has tripped - (DisplaySPRTs 2 NSensors SPRT_Data) (newlme) (display *** ONE OR BOTH OF THE ) (display DName) (display DEVICES, ) (newline) (if (eq? (car SPRT_Data) (* NSensors 2)) (begin

(display SENSOR NUMBER A ) (display NSensors)

(display . OR SENSOR NUMBER Bl MAY BE FAILING

(newline) (newline) ) (begin

(display SENSOR NUMBER A

(if (even' (car SPRT_Data) ) (begin

(display (/ (car SPRT_Data) 2)) (displav OR SENSOR NUMBER B ) (display (+ (/ (car SPRT_Data) 2) 1) ) (display MAY BE FAILING *** ) ) (begin (display (+ (quotient (car SPRT_Data) ) ) (display OR SENSOR NUMBER B ) (display (+ (quotient (car SPRT_Data) 2) 1 ) ) (display MAY BE FAILING *-* ) ) )

(newlme) (newlme) ) ) ) ) ) )

^;; MultipleDevice procedure This procedure contains the logic for the ^;; analysis of a SPRT group which has 3 or more devices, (define MultipleDevice

(lambda (InputData SPRT_Data)

First use temporary variables to extract the data from the InputData list, the number of tripped SPRTs, the names of the tripped SPRTs, and the list of tripped SPRTs for the last device, and two temporary variables for the main do loop in the procedure. ⁽let* ( (NDevices (car InputData) ) (NSensors (cadr InputData)) (DName (caddr InputData) ) (VName (cadddr InputData) ) (NSPRTs (length SPRT Data) ) (DeviceList (NameTr lppedSPRTs NDevices SPRT_Data) (PreviousDevice (car (reverse DeviceList) ) )

(CurrentDevice () ) (NextDevice ()) ) ; ; The logic depends upon the number of tripped SPRTs, (cond ( (eq? NSPRTs (* NSensors NDevices) )

In this case, all of the SPRTs have tripped. First write a message showing current the device and physical variable

(display For the ) (display VName) (display physical variable of the ) (display DName)

(display devices: ) (newline)

(display All ) (display NSPRTs)

(display SPRTs have tripped ) (newline)

(display *** ALL ) (display NDevices) (display OF THE )

(display DName) (display DEVICES HAVE FAILED *** )

(newline) (newline) )

( ( NSPRTs 0)

In this case, 1 or more of the SPRTs have tripped. So we determine if any device or sensors have failed. First write a message showing the current device and physical variable. Then the names of the tripped SPRTs are displayed, (display For the ) (display VName)

(display physical variable of the ) (display DName) (display devices: ) (newlme)

(display These ) (display NSPRTs) (display of the ) (display (* NSensors NDevices)) (display SPRTs have tripped ) (newline) (DisplaySPRTs NDevices NSensors SPRT_Data) (newlme)

A do loop is used to step through the list of names of tripped SPRTs. For each device, we can determine if the device has failed, may be failing, or if one of its sensors has or may be failing The αo list steps from device 1 to NDevices .

(do ( (SubList DeviceList (cdr SubList) ) ) ( (null? SubList) (newline) ) Body of do loop.

First I reset the temporary variables which contain the list of tripped SPRTs for the current and next devices . set! CurrentDevice (car SubList))

If this is the last pass through the do loop, then current device is number NDevices and the next device is the first device [i.e., device number A] . (if (null? (cdr SubList) )

(set! NextDevice (car DeviceList)) (set! NextDevice (cadr SubList)) )

The logic depends upon the number of tripped SPRTs for the current and next devices, (cond ( (and (not (null' CurrentDevice)) (not (null' NextDevice) ) ) In this case, the current device and the next device contain tripped SPRTs. Next we check to see if the tripped SPRTs imply a sensor has failed, (if (and (eq? (length CurrentDevice) 1) (eq? (length NextDevice) 1) (eq? (cadar CurrentDevice) (cadar NextDevice) ) ) In this case, the current and next devices contain 1 tripped SPRT and the sensor num ; ; bers of the SPRTs are the same, (if (eq? NSensors 1)

If the devices contain only l sensor, then either the device or sensor has failed, begin (display *** DEVICE NUMBER ) (display (caar NextDevice) ) (display OF THE ) (display DName) (display DEVICES, ) (newlme) (display OR SENSOR NUMBER ) (display (caar NextDevice) ) (display (cadar CurrentDevice)) (display HAS FAILED *** ) (newline) ) The devices contain more than 1 sensor We can conclude a sensor has failed.

(begin (display *** SENSOR NUMBER (display (caar NextDevice) ) (display (cadar CurrentDevice) ) (display HAS FAILED *** ) (newline) ) ) In this case more than 1 SPRT either of the devices has tripped, therefore we can conclude that the device has failed, (begin

(display *** DEVICE NUMBER ) (display (caar NextDevice) ) (display OF THE ) (display DName) (display DEVICES HAS FAILED *** ) (newlme) ) ) )

Next we check to see if the current device contains tripped SPRTs while the previous and next devices don t . (and ( (length Cu rentDevice) 0) (eq? (length PreviousDevice) 0) (eq? (length NextDevice) 0) ) If only one SPRT has tripped, then we can t conclude a failure, onlv that failure of the current device or a sensor is possible . (if (eq? (length Cur entDevice) 1) (begin

(display *** DEVICE NUMBER ) (display (caar CurrentDevice)) (display OR DEVICE NUMBER ) (display (caddar CurrentDevice)) (display OF THE ⁾ (display DName) (display DEVICES, ) (newlme) (display SENSOR NUMBER ) (display (caar CurrentDevice)) (display (cadar CurrentDevice)) (display , OR SENSOR NUMBER ) (display (caddar Cu rentDevice)) (display (cadar CurrentDevice)¹ (display MAY BE FAILING *** , (newlme) ) In this case, more than 1 SPRT in the current device has tripped. ThereLore we can conclude that the current or the next device has failed.

(begin (display *** DEVICE NUMBER ) (display (caar CurrentDevice)) (display OR DEVICE NUMBER ) (display (caddar CurrentDevice)) (display OF THE ) (display DName) (newline) (display DEVICES HAS FAILED *** ) (newline) ) ) ) ) The last step is to update the PreviousDevice variable.

(set! PreviousDevice CurrentDevice) ) )

NameTr ippedSPRTs pr ocedur e - This inter nal pr ocedur e takes a lis t of tripped SPRTs and converts each number of the tripped SPRT into the SPRT name [e.g., A 1 B 1, is the corresponding SPRT name for tripped SPRT 1] . The output is a list of NDevs elements. Each of the elements contains the name of tripped SPRTs for the corresponding device number [e.g., the 3rd element contains the names of the tripped SPRTs for device C] . If none of the SPRTs for a device have tripped, then the cor esponding element is null. This procedure is applied only to device groups that contain 3 or more devices, (define NameTrippedSPRTs (lambda (NDevs SPRT_List)

; ; Create local list to store output data, (let ( (DevList ()) )

;; A do loop is used to step through each of the devices, from

;; device A to device NDevs. The loop variable, IDev, is the current

; ; device number .

(do ( (IDev 1 (+ IDev 1) ) )

( ( Ndevs IDev) (reverse DevList) ) ; Reverse order of output, (set! DevList

Apply the following unnamed function to each of the elements in SPRT_List. Any null elements are removed from the list returned by map. Finally, the returned list is added to DevList. (cons

(RemoveNull ; Call function to remove nulls,

(map

(lambda (SPRT) ; Unnamed internal function, (if (eq? (modulo SPRT NDevs) (modulo IDev NDevs)) ;; If the current element in SPRT_List is a multiple of IDev, then: (if (eq? (modulo SPRT NDevs) 0)

In this case the devices in the SPRT are device A and device NDevs. Hist

;; Return list containing: letter ref- ; ; ering to first device in the SPRT, (ascii- symbol (+ NDevs 64)) ;; number refering to the sensor, (quotient SPRT NDevs) the letter of the second device, A

;; and the sensor number again, (quotient SPRT NDevs) ) Else, (list

Return list containing: letter ref- ;; ering to first device in the SPRT, (ascii- symbol (+ IDev 64)) number refering to the sensor, (+ (quotient SPRT NDevs) 1) the letter of the second device, (ascii- symbol (+ IDev 65)) ; ; and the sensor number again. (+ (quotient SPRT NDevs) 1) ) ) ) ) SPRT_List ) ) ; ; Add result from RemoveNull to DevList. DevList ) ) ) ) ) )

RemoveNull procedure - This internal procedure recursively removes the null elements from an input list [Lst] . (define RemoveNull (lambda (Lst) ( i f (null? Lst ) I f end of Lst is r eached , 0 return null . ( i f (null ? (car Lst) ) If lst element in Lst is null,

(RemoveNull (cdr Lst) ) ignore it. (cons (car Lst ) Else add lst element to output.

(RemoveNull (cdr Lst)) ) ) ) ) )

DisplaySPRTs Procedure - This internal procedure outputs the names of the tripped SPRTs in Lst. For each tripped SPRT listed in Lst, the procedure determines the name [e.g., A1-A2, or Al-Bl] of the SPRT and writes it to the screen, (define DisplaySPRTs

(lambda (NDevs NSens Lst)

(cond ((null? Lst) #t) ; Lst is empty, don t do anything.

( (eq? NDevs 1) ; Case for single device SPRT.

(if (eq? (car Lst) NSens) (begin

In this case, the last SPRT [AN-A1] has tripped, (display A ) (display NSens) (display -Al )

Display triped SPRTs in the remainder of the list. (DisplaySPRTs NDevs NSens (cdr Lst) ) ) (begin

SPRT number AX-AX+1, where X is the first number in the list, has tripped, (display A ) (display (car Lst) ) (display -A ) (display (+ (car Lst) 1)) (display )

Display tripped SPRTs in the remainder of the list. (DisplaySPRTs NDevs NSens (cdr Lst)) ) ) ) ( (eq? NDevs 2) ; Case for dual device SPRT.

(if (eq? (car Lst) (* NSens 2)) (begin

In this case, the last SPRT [AN-B1] has tripped, (display A ) (display NSens) (display -Bl )

Display tripped SPRTs in the remainder of the list. (DisplaySPRTs NDevs NSens (cdr Lst) ) ) (begin

(display A ) (if (even? (car Lst)) (begin

SPRT number AX-BX+1, where X is the first number ;; in the list divided by 2, has tripped, (display (/ (car Lst) 2)) (display -B ) (display (+ (/ (car Lst) 2) 1)) ) (begin

SPRT number AX-BX, where X is .5 + the first number in the list divided by 2, has tripped, (display (+ (quotient (car Lst) 2) 1)) (display -B )

(display (+ (quotient (car Lst) 2) 1)) ) ) (display )

;; Display tripped SPRTs in the remainder of the list. (DisplaySPRTs NDevs NSens (cdr Lst)) ) ) ) ⁽else ; Case for multiple device SPRT.

(if (eq? (modulo (car Lst) NDevs) 0)

If the tripped SPRT is a multiple of the number of devices, then the SPRT number is NX -AX, where N is the number of devices, and X is the sensor number, (begin

;; Letter representing device N. (display (ascii- symbol (+ NDevs 64))) ; ; Number of the sensor . (display (quotient (car Lst) NDevs) )

Letter representing device A. (display -A ) ; ; Number of the senso . (display (quotient (car Lst) NDevs) ) ) (begin

If the tripped SPRT is not a multiple of the numoer of devices, then the number of the first device is given by the remainder of SPRT/NDevs .

Letter representing first device in tripped SPRT. (display (ascii- symbol (+ (modulo (car Lst) NDevs) 64)))

Number of the sensor . (display (+ (quotient (car Lst) NDevs) 1) ) (display - )

Letter representing second device in tripped SPRT (display (ascii- symbol (+ (modulo (car Lst) NDevs) 65))) ; ; Number of the sensor .

(display (+ (quotient (car Lst) NDevs) 1) ) ) ) (display )

;; Display tripped SPRTs in the remainder of the list. (DisplaySPRTs NDevs NSens (cdr Lst) ) ) ) ) )

Main body of Analyze procedure.

The Analyze procedure is passed input data and tripped SPRT data super lists. Each element of these lists contains the input data and lis_t of tripped SPRTs for a corresponding group of independent SPRTs. This portion of the procedure is a control routine which each element of the super lists to the AnalyzeGroup procedure. f (null? SPRT_Lιst) Quit if end of list is reached. #t (begin

(AnalyzeGroup (car GroupList) Analyze lst element of the lists. (car SPRT_Lιst) GroupCount) (Analyze (cdr GroupList) Analyze remainder of the lists. (cdr SPRT_Lιst) (+ 1 GroupCount) ) ) ) ⁾

End of the Analyze pr ocedur e .

Claims

What Is Claimed Is:

1. A method of testing both an industrial process and a sensor for deteπnining fault conditions therein, comprising the steps of: deteπnining automatically, using computer means, a configuration of a n inimum number of sensor pairs needed to test the industrial process and the sensor for state of operation; operating at least a first and second sensor to form at least one sensor pair to redundantly detect at least one physical variable of the industrial process to provide a first signal from said first sensor and a second signal from said second sensor, each said signal being characteristic of the one physical variable; obtaining a difference function characteristic of the arithmetic difference pairwise between said first signal and said second signal at each of a plurality of different times of sensing the one physical variable; obtaining a frequency domain transformation of said first difference function to procure Fourier coefficients corresponding to Fourier frequencies; generating a composite function over time domain using the Fourier coefficients; obtaining a residual function over time by determining the arithmetic difference between the difference function and the composite function, the residual function being substantially free of serially correlated noise; operating on the residual function using the computer means for performing a statistical analysis technique to determine whether an alarm condition is present in at least one of the industrial process and the sensor, the residual function including white noise characteristics of an uncorrelated function of reduced skewness relative to the difference function and being input to the statistical analysis technique; and said at least one sensor pair providing alarm information to an operator of the industrial process allowing modification of at least one of the industrial process and said at least first and second sensor when an alarm condition is detected.

2. The method described is Claim 1 wherein said computer means comprises an artificial intelligence system.

3. The method as defined in Claim 1 wherein the residual function comprises reduced Markov dependent noise.

4. The method as defined in Claim 1 wherein the industrial process comprises at least one of a chemical process, a mechanical process and an electrical operational process.

5. The method as defined in Claim 1 wherein the step of obtaining Fourier coefficients comprise iteratively determining the minimum number of Fourier harmonics able to generate the composite function,

6. The method as defined in Claim 1 further including at least one of the steps of modifying the industrial process or changing the sensor responsive to the alarm condition.

7. A method of testing both an industrial process and a sensor for determining fault conditions therein, comprising the steps of: determining, using computer means, a configuration of a minimum number of sensor pair signals needed to characterize the industrial process and the sensor state of operation; operating at least a first sensor to detect at least one physical variable of the industrial process to provide a real signal from said first sensor; generating an artificial signal characteristic of the one physical variable; forming a sensor pair signal from said real signal and said artificial signal; obtaining a difference function characteristic of the difference pairwise between said real signal and said artificial signal at each of a plurality of different times of sensing the one physical variable; obtaining a frequency domain transformation of said difference function; generating a composite function over a time domain; obtaining a residual function over time by determining the difference between the frequency transformed difference function and the composite function; operating on the residual function using the computer means for performing a statistical analysis technique to determine whether an alarm condition is present in at least one of the industrial process and the first sensor, the residual function including white noise characteristics of an uncorrelated signal of reduced skewness relative to the difference function and being input to the statistical analysis technique; and said first sensor providing alarm information to an operator of the industrial process allowing modification of at least one of the industrial process and the first sensor when an alarm condition is detected.

8. The method as defined in Claim 7 wherein the step of obtaining a frequency domain transformation comprises performing a Fourier transformation.

9. The method as defined in Claim 7 wherein the steps of obtaining a composite function over time comprises performing an auto regressive moving average analysis.

10. The method as defined in Claim 7 further including the step of determining a difference function for both the artificial signal and the real signal, as well as a separate pair of real signals.

11. The method as defined in Claim 7 wherein the residual function comprises reduced Markov dependent noise.

12. The method as defined in Claim 8 wherein the step of obtaining a frequency domain transformation comprises obtaining Fourier coefficients iteratively to determine the minimum number of Fourier harmonics able to generate the composite function.

13. A system for automatically configuring sensors for testing both an industrial process and a sensor for determining a fault condition therein, comprising: computer means for automatically configuring a minimum number of sensor pair signals needed to characterize the industrial process and the sensor state of operation; at least a first sensor to detect at least one physical variable of the industrial process to provide a first signal from said first sensor; first means for generating a second sensor signal for comparison with said first signal from said first sensor; second means for deterrmning a difference function characteristic of the arithmetic difference pairwise between said first signal and said second signal at each of a plurality of different times of sensing the one physical variable; third means for obtaining a residual function over time by means for determining the arithmetic difference between the difference function and the composite function, the residual function including white noise characteristics of an uncorrelated signal of reduced skewness; fourth means for operating on the residual function, said fourth means including the computer means for executing a computer program for performing a statistical analysis technique and for deterniining whether an alarm condition is present in at least one of the industrial process and the sensor and with said second means, said third means, and said fourth means cooperatively providing a function comprised of said white noise characteristics of uncorrelated signal of reduced skewness relative to the difference function as an input to the statistical analysis technique; and means for providing information allowing modification of at least one of the industrial process and the sensor when an alarm condition is detected.

14. The system as defined in Claim 13 further including means for obtaining a frequency domain transformation of said difference function.

15. The system as defined in Claim 13 wherein said computer means comprises an artificial intelligence system.

16. The system as defined in Claim 13 wherein said means for generating a second signal comprises the computer means for executing a computer program.

17. The system as defined in Claim 16 wherein the computer program includes an autoregressive moving average procedure.

18. The system as defined in Claim 13 wherein the system includes at least one pair of sensors for detecting each of the physical variables.

19. The system as defined in Claim 13 wherein said computer means executes a computer program including a statistical probability ratio test on the residual function.

20. The system as defined in Claim 13 further including means for changing at least one of the industrial process and substituting another sensor for a defective sensor.